Sym metry , I n tegra b i l i ty a nd Geometry : Methods a nd App l i ca t i ons S I GMA 6 ( 20 1 0 ) , 76 , 45 pa ges Erlangen Program at Large - 1 : Geometry of Invariants Vladimir V . KISIL School of Mathematics , University of Leeds , Leeds LS 2 9 JT , UK E - mail : k is i l v @ maths.leeds. ac.uk URL : h t tp : / / www . mat hs . l e eds . ac . uk / ∼ k i s i lv / Received April 20 , 201 0 , in final form September 1 0 , 20 1 0 ; Published online September 26 , 20 1 0 doi:10.3842/SIG MA .2010.076 Abstract . This paper presents geometrical foundation for a systematic treatment of three main ( elliptic , parabolic and hyperbolic ) types of analytic function theories based on the representation theory of SL2(R) group . We describe here geometries of corresponding do - mains . The principal ro ˆ le is played by Clifford algebras of matching types . In this paper we also generalise the Fillmore – Springer – Cnops construction which describes cycles as points in the extended space . This allows to consider many algebraic and geometric invariants of cycles within the Erlangen program approach . Key words : analytic function theory ; semisimple groups ; elliptic ; parabolic ; hyperbolic ; Clifford algebras ; complex numbers ; dual numbers ; double numbers ; split - complex numbers ; Mo ¨ bius transformations 201 0 Mathematics Subject Classification : 30 G 35 ; 22 E 46 ; 30 F 45 ; 32 F 45 Contents 1 I n t r o d u ct io n 2 1 . 1 Backgr o u n d a n d h i s t o r y ...... 2 1 . 2 H igh l ight s o f o b t ai n ed r e s u l t s ...... 3 1 . 3 T he p a p e r o utlin e ...... 4 2 E l l ipt ic , pa r abo l ic and hyp erbolic h o m ogen eous spa c e s 5 2.1 SL2( R) gr o up a n d Cli f for d alg e br as ...... 5 2 . 2 A ctio n s o f s u b gr o u p s ...... 6 2 . 3 I n v a r ian c e o f c y c l e s ...... 1 0 3 Space of cy cle s 1 1 3 . 1 F i l l m o r e – Sprin g e r – C n o p s c o n s t r u c t i o n ( F S C c ) ...... 11 3 . 2 F ir st in v ari a nt s o f cy c l e s ...... 14 4 J oi n t i n v aria nt s : o r t hogo na l i ty a nd inv e r s ions 16 4 . 1 In v a r ian t o r t hog o n ali t y ty p e c o n di t ion s ...... 16 4.2 Inversions in cycles ...... 1 9 4 . 3 F oc al o rt hog on a lit y ...... 2 1 5 M et ri c pro per t ies f r o m cy c l e inva ria n t s 24 5 . 1 Dista n c es a n d len gt hs ...... 24 Sym metry comma I n tegra b i l i ty a nd Geometry : .... Methods a nd App l i ca t i ons .... S I GMA 6 open parenthesis 20 1 0 closing \noindent Sym metry , I n tegra b i l i ty a nd Geometry : \ h f i l l Methods a nd App l i ca t i ons \ h f i l l S IGMA6 ( 20 1 0 ) , 76 , 45 pa ges parenthesis5 comma . 2 76 C commao n 45f o pa r ges mal p r op e r ti e s o f M o¨ bi u s m a p s . . Erlangen...... Program .... at .... Large hyphen 1 : .... Geometry .... of Invariants \noindentVladimir. 2 VErlangen 7 period 5 . 3 KISIL\ Ph f i ll e rProgram p e n d\ ich u f i la l l rityat a\ h f i n l l Large d o r− t ho1 go : \ nh f i l a l lGeometry it y .\ h . f i l l . of Invariants School...... of Mathematics comma .. University of Leeds comma Leeds LS 2 9 JT comma .. UK \noindentE hyphen. . mailVladimir 29 : .. k is V .. .i l KISIL v .. at .. m a t h s period l e e d s period .. a c period u k URL : .. h t tp : slash slash www period mat hs period l e eds period ac period uk slash thicksim k i s i lv slash \noindentReceived AprilSchool 20 comma of Mathematics 201 0 comma in , final\quad formUniversity September 1 0of comma Leeds 20 ,1 0 Leeds semicolon LS Published 2 9 JT , online\quad SeptemberUK 26 comma 20 1 0 E d− omail i : 1 0 :period\quad 3 8k 4 2 i sslash\quad SI G ..i M lA v ..\quad period 2$ 0 @ 1 0 $ period\quad 0 7maths.leeds. 6 \quad a c . u k Abstract period This paper presents geometrical foundation for a systematic treatment of three \noindentmain openURL parenthesis : \quad ellipticht comma tp : parabolic //www. and mathshyperbolic . closing l e eds parenthesis . ac .types uk of $/ analytic\sim function$ theories k i s based i l v on / the representation theory of SL sub 2 open parenthesis R closing parenthesis group period .. We describe here geometries of corresponding do hyphen\noindent Received April 20 , 201 0 , in final form September 1 0 , 20 1 0 ; Published online September 26 , 20 1 0 mains period The principal r circumflex-o le is played by Clifford algebras of matching types period In this paper we \ centerlinealso generalise{doi:10.3842/SIG the Fillmore endash Springer endash Cnops\quad constructionMA \quad which. 2 describes 0 1 0 .cycles 0 7 as 6 points} in the extended space period .. This allows to consider many algebraic and geometric invariants of \ centerlinecycles within{ Abstract the Erlangen . program This paper approach presents period geometrical foundation for a systematic treatment of three } Key words : .. analytic function theory semicolon .. semisimple groups semicolon .. elliptic semicolon .. parabolic semicolon .. hyperbolic semicolon\ centerline {main ( elliptic , parabolic and hyperbolic ) types of analytic function theories based on the } Clifford algebras semicolon complex numbers semicolon dual numbers semicolon double numbers semicolon split hyphen complex numbers semicolon\ centerline { representation theory of $ SL { 2 } ( R ) $ group . \quad We describe here geometries of corresponding do − } M dieresis-o bius transformations \ centerline201 0 Mathematics{mains Subject . The Classification principal : r 30 G $ 35\hat semicolon{o} $ 22 le E 46 is semicolon played 30 by F 45 Clifford semicolon algebras 32 F 45 of matching types . 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V . Makareviq \ centerlineA period1 V Introduction{ periodI pr Makareviq $ya$ miznu tetiv yloma $yu , $ } 1 ..This Introduction paper describes geometry of simples two - dimensional domains in the spirit of the Er- \ centerlineThislangen paper describes{ i lukgeometry sgibaets of simples $ ya $two v hyphen krug dimensional\quad ... domains} in the spirit of the Erlangen programprogram of F period of F . .. Klein Klein influenced influenced by works by works of S period of S . Lie Lie comma , for .. itfor s it development s development .. see books books .. open [ square bracket 3 comma ..\ centerline 6 7 comma3 ,6 ..{ 7 6A 5, . closing V 6 .5] squareMakareviq and bracket their} references . Further works in this series will use the Erlangen andapproach their references for analytic period .. Further works in this series will use the Erlangen approach for analytic \noindentfunctionfunction theories1 \ theoriesquad andIntroduction spectral and spectral theory of theory operators of .. operators open square bracket[ 5 0 ] .5 0 closing In the square present bracket paper period we .. are In the present paper we are focusedfocused on the geometry and study objects in a plane and their properties which are invariant \noindenton theunder geometryThis and paper study describes objects in a geometry plane and their of simples properties two which− aredimensional invariant under domains in the spirit of the Erlangen linearlinear hyphen - fractional fractional transformations transformations associated associated t o t the o the SL subSL2 (R open) group parenthesis . The R closing basic observation parenthesis group period .. The basic observation\noindentis that isprogram that geometries of F obtained . \quad inKlein this way influenced are naturally by works classified of S as .e Lie l liptic , \quad, parabolicfor itand s development \quad see books \quad [ 3 , \quad 6 7 , \quad 6 5 ] andgeometrieshyperbolic their obtained references. in this . way\quad are naturallyFurther classified works as in e l this liptic comma series parabolic will use and hyperbolic the Erlangen period approach for analytic 1 period1 . 1 1 .. Background Background and history and history \noindentWeWe repeatedly repeatedlyfunction meet such meet theories a suchdivision a and division of various spectral of mathematical various theory mathematical obj of ects operators into three obj ects main\quad into classes[ three 5 period 0 ] main . \quad classesIn . the present paper we are focused onThey theThey are geometry named are named by and the by historically study the historically objects first example in first a endash example plane the and –classification the their classification properties of conic sections of conic which : ..sections ellip are hyphen invariant : ellip under t ic- comma t ic , parabolic parabolic and and hyperbolic hyperbolic endash – however however the the pattern pattern persistently persistently reproduces reproduces itself in itself many in very many \noindentdifferentvery areas differentl i n e open a r − parenthesis areasfractional ( equations equations transformations , comma quadratic quadratic forms associated forms , metrics comma t , metrics manifolds o the comma $ , SL operators manifolds{ 2 } comma ,( etc . R operators ) ) $ comma group etc . period\quad The basic observation is that closinggeometries. parenthesis We obtained will period abbre .. We in - viatewill this abbre this way hyphen separation are naturally as EPH classified - c lassification as e l. liptic The common , parabolic origin and hyperbolic . viateof this this separation fundamental as EPH division hyphen can c lassification b e seen period from .. the The simple common picture origin of of thisa coordinate fundamental line division split can by \noindentb ethe seen zero from1 . into the 1 simple\ negativequad pictureBackground and of positive a coordinate and half history line - axes split by : the zero into negative and positive half hyphen axes : ( 1 . 1 ) \noindentopen parenthesisConnectionsWe repeatedly 1 period between 1 closing meet different parenthesis such aobj division ects admitting of various EPH - mathematical classification are obj not ects limited into t three o main classes . TheyConnectionsthis are common named between by source different the historically . obj ects There admitting are first many EPH example hyphen deep results classification−− the linking classification are , not for limited example t o of this , ellipticity conic sections of : \quad e l l i p − tcommon icquadratic , parabolic source forms period and , .. metricsThere hyperbolic are and many operators−− deephowever results . linking the On thecomma pattern other for examplepersistentlyhand there comma are ellipticity reproduces still a lot of quadratic of itselfwhite forms in commamany very differentmetricsspots and and areas operators obscure ( period equations gaps .. On b etween the , quadratic other some hand subj there forms ects are still as, metrics well a lot . of white , manifolds spots and obscure , operators gaps , etc . ) . \quad We will abbre − viateb etween thisFor some example separation subj ects , it as i s well as well EPH period known− c that lassification elliptic operators . \quad are effectivelyThe common treated origin through of this complex fundamental division can bFor eanalysis seen example from comma , which the it can simple i s well be knownnaturally picture that identified elliptic of a coordinate operators as the aree l effectively lipticline analytic split treated by function through the zero complex theory into[ 3 negative 7 , 4 and positive h aanalysis l f 1− ] .axes comma Thus : which there can i be s a naturally natural identified quest for ashyperbolic the e l lipticand analyticparabolic functionanalytic theory open function square theories bracket 3 7 comma .. 4 1 closing square, bracket which period will b .. e Thus of similar importance for corresponding types of operators . A search for \ hspacetherehyperbolic i∗{\ s a naturalf i l l } function( quest 1 . for 1 theory )hyperbolic and parabolic analytic function theories comma which will b e of similarwas importance attempted for several corresponding times typesstarting of operators from 1 930period ’ s .. , A see search for forexample hyperbolic [ 7 0function , 5 theory 6 , 5 9 ] . ConnectionswasDespite attempted ofbetween several some times important different starting advances fromobj 1ects 930 the quoteright admitting obtained s comma hyperbolic EPH − seeclassification for theory example does open not square are look not bracket as limited natural 7 0 comma t o .. this 5 6 comma .. 5 9 closingcommonand square source complete bracket . period as\quad ..There Despite are of some many deep results linking , for example , ellipticity of quadratic forms , metricsimportant and advances operators the obtained . \quad hyperbolicOn the theory other does hand not look there as natural are and still complete a lot as of white spots and obscure gaps b etween some subj ects as well .

For example , it i s well known that elliptic operators are effectively treated through complex analysis , which can be naturally identified as the e l liptic analytic function theory [ 3 7 , \quad 4 1 ] . \quad Thus there i s a natural quest for hyperbolic and parabolic analytic function theories , which will b e of similar importance for corresponding types of operators . \quad A search for hyperbolic function theory

\noindent was attempted several times starting from 1 930 ’ s , see for example [ 7 0 , \quad 5 6 , \quad 5 9 ] . \quad Despite of some important advances the obtained hyperbolic theory does not look as natural and complete as Erlangen Program at Large hyphen 1 : .... Geometry of Invariants .... 3 \noindenthlineErlangenErlangen Program Program at Large at - Large 1 :− 1 : \ h f i l l GeometryGeometry of of Invariants Invariants \ h f i l l 3 3 complex analysis i s period .. Parabolic geometry was considered in an excellent book open square bracket 7 2 closing square bracket comma which\ [ \ r i u s l stille {3em}{0.4 pt }\ ] a source of valuable inspirations period .. However the corresp onding quotedblleft parabolic calculus quotedblright .. described in variouscomplex places analysis open square i s .bracket Parabolic 1 1 comma geometry 2 .. 3 comma was considered .. 7 3 closing in square an excellent bracket i books rather [ 7 trivial 2 ] , whichperiod \noindentTherei s i still s alsocomplex a a source recent analysis interest of valuable to thisi s inspirations t . opic\quad in differentParabolic . areasHowever : geometry.. differential the corresp was geometry onding considered open “ parabolic square in bracket an calculus excellent 7 comma .. book 1 1 comma [ 7 2 ] , which i s still a1 source ..” 2 comma described of 2 valuable .. 1 comma in various inspirations.. 8 comma places 2 closing [ 1 . 1\ , squarequad 2 3 bracketHowever , 7 comma 3 ] the i s rathermodal corresp logic trivial ondingopen . square ‘‘ bracket parabolic 5 .. 4 closing calculus square ’’ bracket\quad commadescribed in quantumvarious mechanicsThere places i s open also [ 1 square a 1 recent , 2 bracket\quad interest 2 ..3 8 to , comma\ thisquad t 2 opic7 .. 3 9 incomma ] different i s .. rather 5 7 areas comma trivial : .. 4 differential 9 closing. square geometry bracket [ comma7 space hyphen t ime geometry, open 1 1 square , 1 bracket 2 , 2 1 1 .. , 0 comma 8 , 2 2] , .. modal 5 comma logic [ 5 4 ] , quantum mechanics [ 2 8 , 2 9 , There2 ..5 6 i 7 comma s, also 4 .. 9 5 ]a 8, recent spacecomma - 2 t interest .. ime 4 comma geometry to 2 2 this comma [ 1 t .. 0 opic 6 , 12 closing in 5 ,different square 2 6 bracket , areas 5 8comma , 2 : \ hypercomplexquad 4 , 2 2differential , 6 analysis 1 ] , open geometry square bracket [ 7 , 1\ 3quad 1 1 , comma1 \quadhypercomplex .. 1 92 comma , 2 \ 2quad .. analysis 0 closing1 , [\ squarequad 1 3 ,8 bracket 1 , 9 2 , period ] 2 , modal 0 .. ] A . brief logic A history brief [ 5of history the\quad t opic of4 the can ] , tb quantumopice found can b mechanics e found [ 2 \quad 8 , 2 \quad 9 , \quad 5 7 , \quad 4 9 ] , space − t ime geometry [ 1 \quad 0 , 2 \quad 5 , 2 in\quad openin [ square16 2 , ] and\ bracketquad further5 1 28 closing references, 2 \ squarequad are4 bracket , provided 2 and2 , further\ inquad the references6 above 1 ] papers, are hypercomplex provided . in the above analysis papers [ period 1 3 , \quad 1 9 , 2 \quad 0 ] . \quad A brief history of the t opic can b e found inMost [ 1 ofMost 2 previous ] and of previous research further had research references an algebraic had an flavour are algebraic provided period flavour .. An in alt . the ernative An above alt approach ernative papers t o approach analytic . func t o hyphen analytic t ionfunc theories - t ion based theories on the based representation on the representationtheory of semisimple theory Liegroups of semisimple was developed Lie groups in the was developed Mostseriesin of of the previous papers series open of research square papers bracket [ 3 had 3 , 3 an 3 comma 3 algebraic 5 , .. 3 3 5 comma 6 flavour , 3 .. 7 3 ,.. . 6\ comma 3quad 8 ,An .. 4 3 0alt 7 , comma ernative 4 1 .. ] 3. 8 comma Particularly approach .. 4 0 comma t o analytic .. 4 1 closing func square− brackett ion, period some theories .. elements Particularly based ofcommahyperbolic on the some representation function elements of theory hyperbolic theory were function builtof semisimple in [ 3 5 , Lie 3 7 groups ] along was the developed same in the seriestheorylines were of as papers built the elliptic in open [ 3 square one3 , –\quad bracket standard3 3 5 5 complex comma , \quad .. analy33 7\ closingquad - sis square6 . , \quad bracket Covariant3 along 7 , functional the\quad same3 lines calculus8, as\ thequad ellipticof 4 0 one , \ endashquad standard4 1 ] . \quad Particularly , some elements of hyperbolic function complextheoryoperators analy were hyphen built and respective in [ 3 5 covariant , \quad spectra3 7 ] were along considered the same in lines [ 3 4 as, the 4 3 ] elliptic . one −− standard complex analy − s isis s period . \Thisquad .. CovariantpaperCovariant continues functional functional this calculus line of of calculus research operators and and of respectivesignificantly operators covariant and expands respective spectra results were considered covariant of the earlier spectra were considered inin [ openpaper 3 4 square , [ 5\quad 1 bracket ] , see4 3 3also 4] comma . [ 4 .. 4 5 3 ] closing for an square easy - bracket reading period introduction . A brief outline of the ThisErlangen paper continues Pro - this gramme line of at research Large and , which significantly includes expands geometry results , analytic.. of the .. functions earlier and functional Thispapercalculus paper open continuessquare , i s written bracket this 5 in 1 closing [ line 5 0 ] square of . research bracket comma and significantly see also open square expands bracket 4 results .. 5 closing\quad squareo fbracket the \ forquad an easye a r hyphen l i e r readingpaper1 introduction . [ 2 5 1 Highlights] ,period see .. also A ofbrief [ obtained outline4 \quad of the results5 ]Erlangen for an Pro easy hyphen− reading introduction . \quad A brief outline of the Erlangen Pro − grammegrammeIn the at at Large previousLarge comma , paperwhich which includes includes [ 5 1 ] geometry geometry we identify comma , analytic analytic geometric functions functions obj ects and functional called and functional calculuscycles comma calculus[ 7 i 2 s written ] , i s written inin [ open, 5 0 squarewhich ] . bracket are circles 5 0 closing , parabolas square bracket and hyperbolas period in the corresponding EPH cases . They 1 periodare invariants 2 .. Highlights of the of obtained M o¨ bius results transformations , i . e . the natural geometric objects in the \noindentIn thesense previous1 of . the 2 paper Erlangen\quad .. openHighlights program square bracket . of obtained 5 Note 1 closing also resultssquare that bracket cycles are.. we algebraically identify geometric defined obj ects through called .. cycles .. open square bracketthe 7 2 quadratic closing square expressions bracket comma ( 2 ... 1which 0 b are ) which circles may comma lead t o interesting connections with the \noindentparabolasinnovative andIn thehyperbolas approach previous in t theo the paper corresponding geometry\quad presented EPH[ 5 cases 1 ] inperiod\quad [ 7 1 .. ]we They . In identify are this invariants paper geometric of we the M systematically dieresis-o obj ects bius called \quad c y c l e s \quad [ 7 2 ] , \quad which are circles , parabolastransformationsstudy those and comma hyperbolas cycles i period throughin e period the the corresponding naturalan essential geometric EPH extension objects cases in the . \ sensequad of the ofThey the Fillmore Erlangen are invariants – program Springer period – of .. the Note M $ \ddot{o} $ biusalsoCnops that cycles construction are algebraically [ 1 defined 6 , through 6 3 ] the abbreviated quadratic expressions in this paper open parenthesis as FSCc 2 . period The 1 0 idea b closing b parenthesis which may transformationsleadehind t o interesting FSCc is connections , to i consider . e . with the cycles the natural innovative not as geometriclo approach ci of points t o objects the from geometry the in initial the presented sensepoint in open ofspace the squarebut Erlangen bracket rather 7 program 1 closing square . \quad bracketNote periodalsoas that points cycles of the are new algebraicallycycle space , see defined Section through 3 . 1 . the Then quadratic many geometrical expressions properties ( 2 . of 1 0 b ) which may leadIn thisthe t .. o point paper interesting spacewe .. systematically may connections b e better .. study expressed with those the .. cycles through innovative .. through properties approach.. an .. of essential the t cycle o extension the space geometry .. . of the Notably presented in [ 7 1 ] . InFillmore tM h i so¨ \bius endashquad linear Springerpaper - fractional we endash\quad Cnops transformationssystematically construction .. of open the\quad square pointstudy bracket space those are 1 6 commalinearised\quad .. 6c yin 3 c closing lthe e s cycle\quad square spacethrough bracket ..\ abbreviatedquad an \ inquad this essential extension \quad o f the paperFillmore as, see FSCc Proposition−− periodSpringer .. The 3 .idea−− 3 .Cnops An interesting construction feature\quad of the[ corresp 1 6 , ondence\quad 6 b 3 etween ] \quad theabbreviated point and in this paper as FSCc . \quad The idea bb ehind ehindcycle FSCc FSCc spaces is is to is consider to that consider many cycles not relations cycles as lo ci notbetween of points as lo from cycles ci the of , initial points which point are from space of but thelocal rather initial nature as in point the cycle space but rather as pointspointsspace of of the , the new looks new cycle like cycle space non comma space - lo cal see , ifSection see Section 3 period 31 period . 1 . ..\ Thenquad manyThen geometrical many geometrical properties of the properties point of the point spacespacetranslated may b b e e better back better expressed t o the expressed point through space propertiesthrough , see for of properties example the cycle space non of - period local the cycle ..character Notably space M of dieresis-o cycle . \quad orthogonality biusNotably linear hyphen M $ \ddot{o} $ bius linear − fractionalfractionalin Figs transformations transformations . 8 and of 1 the 1 point . of spacethe Such point are a linearised non space - point in are the behaviour cycle linearised space comma i s in oft see the enly Proposition cycle thought space 3 to period b , e 3see a period Proposition 3 . 3 . 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Introducing\quad butSuch appears parabolic a non here− withinobjpoint ects the behaviour on Erlangen a common program i s ground oft approach enly with open thought elliptic square and to bracket b e a3 .. characteristic 9 comma .. 4 2 property closingo f nonhyperbolic square− commutative bracket ones period we geometryshould warn but against appears some here common within prejudices the Erlangen suggested program by picture approach ( 1 . 1 [ 3 \quad 9 , \quad 4 2 ] . ThisThis): also demonstrates demonstrates that our that results our cannot results b e reduced cannot t o b the e ordinary reduced differential t o the geometry ordinary differential geometry oror nine nine1 Cayley Cayley. Theendash−− parabolic Klein geometries geometries case i sof unimportant the ofplane the open plane (square has [ “ bracket 7 zero 2 , measure7 2Appendix comma ” Appendix ) C in ] comparison , C [ closing 6 2 ] square t . o the bracket comma open square bracketelliptic 6 2 closing and square hyperbolic bracket onesperiod . As we shall see ( e . g . Remarks 8 . 6 and 5 . 1 1 . 2 ) some \noindentRemarkgeometrical 1Remark period 1features period 1 . 1 .. .are Introducing\quad richerIntroducing in parabolic parabolic obj ectscase parabolic on . a common obj ground ects onwith a elliptic common and ground hyperbolic with elliptic and hyperbolic onesones we we2 should . should The warn warn parabolic against against some case common isome s a limiting common prejudices situation prejudices suggested ( a by contraction suggestedpicture open ) parenthesisby or an picture intermediate 1 period ( 1 .1 position closing 1 ) : parenthesis : 1 periodbe - .. tween The parabolic the elliptic case i and s unimportant hyperbolic open ones parenthesis : all has properties quotedblleft of zero the measure former quotedblrightcan be guessed closing or parenthesis in comparison t1 o . the\quadob elliptic - tainedThe and parabolic as a limit orcase an averagei s unimportant from the latter ( has two ‘‘ . zero Particularly measure ’’ this ) point in comparison of view i s t o the elliptic and hyperbolichyperbolicimplic ones - ones it period ly supposed . \ ..quad As weAs in shall [ we 5 see 6 shall ] open . parenthesis Although see ( e e there. period g . are gRemarks period few confirmations Remarks 8 . 6 8 and period 5of 6 .this and 1 (15 eperiod . . 2 g . ) 1 Fig 1 some period geometrical 2 closing parenthesis features someare geometrical. richer17(E) – in( featuresH)) parabolic, we shall case see . 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Fig $ . 1 7it (ly supposed E ) in $ open−− square$ ( bracket H 5 ) 6 closing ) square , $ bracket period .. Although there are few confirmations of this open parenthesis e periodwe \quad g periods h Fig a l l period\quad 1 7see open\ parenthesisquad ( e E . closing g . \ parenthesisquad Remark endash\quad open parenthesis5 . 2 3 H ) closing\quad parenthesisthat \quad closingsome parenthesis properties comma\quad o f the \quad p a r a b o l i c \quad case \quad cannot \quad b e straightforwardlywe .. shall .. see .. open guessed parenthesis from e period a combination g period .. Remark of the .. 5elliptic period 2 3 and closing hyperbolic parenthesis.. cases that .. . some properties .. of the .. parabolic .. case .. cannot .. b e straightforwardly guessed from a combination of the elliptic and hyperbolic cases period 4 .... V period V period Kisil \noindenthline4 4 \ h f i l l V . V . K i s i l V . V . Kisil 3 period .. All three EPH cases are even less disj oint than it i s usually thought period .. For example comma there \ [ are\ r umeaningful l e {3em}{ notions0.4 pt }\ of centre] of a parabola open parenthesis 2 period 1 2 closing parenthesis or focus of a circle open parenthesis 3 period 1 0 closing parenthesis period 4 period3 ... A open All three parenthesis EPH co cases hyphen are closing even less parenthesis disj oint invariant than it geometry i s usually i s b thought elieved to . b e quotedblleft For example coordinate free quotedblright which3 . \ sometimesquad, thereAll are is three pushed meaningful EPH t o cases notions are of centre even less of a parabola disj oint( 2 than . 1 2 it ) or ifocus s usually of a circle thought( 3 . . 1\ 0quad For example , there arean absolute). meaningful mantra notions period .. However of centre our study of a within parabola the Erlangen ( 2 . program 1 2 ) frameworkor focus reveals of a circle ( 3 . 1 0 ) . two useful4 . notions A ( ..co open - ) invariant parenthesis geometry Definitions i 2 s period b elieved 1 2 and to b .. e open “ coordinate parenthesis free 3 period ” which 1 0 closing sometimes parenthesis closing parenthesis .. mentioned4 . \quadis pushed aboveA ( which cot o− are an) defined invariant absolute by mantra geometry . i However s b elieved our study to b e within ‘‘ coordinate the Erlangen free program ’’ which sometimes is pushed t o ancoordinate absoluteframework expressions mantra reveals and . two\ lookquad useful veryHowever .. notions quotedblleft our ( study non Definitions hyphen within invariant 2 . the 1 2 quotedblright Erlangen and ( 3 program . on 1 0the ) first ) framework glance mentioned period reveals twoAnabove amazinguseful which aspect notions are of this defined\quad t opic by( i s Definitions acoordinate transparent expressions similarity 2 . 1 2 between and and\ lookquad all three very( EPH 3 . “ cases 1 non 0 which) - invariant ) \quad ”mentioned on the above which are defined by coordinatei s combinedfirst glance withexpressions . some non hyphen and look trivial very exceptions\quad like‘ ‘non non hyphen− invariant invariance of ’’ the on upper the half first hyphen glance plane in . the hyperbolicAn caseamazing .. open aspect parenthesis of this Subsection t opic i .. s 7 a period transparent 2 closing similarityparenthesis between.. or non hyphen all three symmetric EPH cases length and orthogonality in the Anparabolic amazingwhich iaspect s combined of this with t some opic non i s- trivial a transparent exceptions similarity like non - invariance between allof the three upper EPH half cases which icase s- combined .. plane open in parenthesis thewith hyperbolic some Lemma non 5− case periodtrivial 22 ( Subsection period exceptions open parenthesis 7 like . 2 ) non p closing or− noninvariance parenthesis - symmetric closing of thelength parenthesis upper and half period− ..plane The elliptic in the case seemshyperbolic torthogonality o be free case from any in\quad the such parabolic( irregularities Subsection .. only\quad 7 . 2 ) \quad or non − symmetric length and orthogonality in the parabolic b ecausecase it is ( the Lemma standard 5 model. 22 by. 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SectionSection 2 describ 2 describ es the SL es sub the 2 openSL parenthesis2(R) group R , closing its one parenthesis - dimensional group comma subgroups its one hyphen and corresponding dimensional subgroups and corresponding homo\noindent hyphenhomo1 - geneous . 3 \quad spacesThe . paper Here corresponding outline Clifford algebras show their relevance and cycles geneousnaturally spaces periodappear Here as correspondingSL2(R)− invariant Clifford obj algebras ects show. their relevance and cycles naturally \noindentappearTo as SLSection study sub 2 cycles open 2 describ parenthesis we extend es R inthe closing Section $ parenthesis SL { 32 the hyphen} Fillmore( invariant R – )$ Springer obj ects group period – Cnops , its construction one − dimensional ( subgroups and corresponding homo − geneousTo studyFSCc spacescycles ) t o we include .extend Here parabolic in corresponding Section .. case 3 the . Fillmore Clifford We also endash refine algebras Springer FSCc endash showfrom Cnops a their traditional construction relevance severe open and restriction parenthesis cycles FSCc naturally closing parenthesis appeart othat include as space parabolic $ SL of cycles{ case2 period} posses(R) .. We the also same refine− metric$ FSCc invariant from as the a traditional initial objects point severe . space restriction . that Cycles space b ecame of cyclespoints posses in a the bigger same metric space as and the got initial their point presentation space period .. by Cycles matrix b ecame . points We derive in a bigger first SL2(R)− Tospace studyinvariants and cycles got their of we cycles presentation extend from in by the matrixSection classic period matrix\quad .. We invariants3 derive the Fillmore first . SL sub−− 2 openSpringer parenthesis−− Cnops R closing construction parenthesis hyphen ( FSCc invariants ) of cyclest o from includeMutual parabolic disp osition case of two . \ cyclesquad We may also be also refine characterised FSCc from through a traditional an invariant severe notions restriction of that space ofthe cycles( classic normal matrix posses and invariants focal the ) same period orthogonalities metric as the , initial see Section point space4 , both . \quad are definedCycles in b matrix ecame pointst in a bigger spaceMutualerms and disp of got FSCc osition their . of Orthogonality two presentation cycles may be in also generalised by characterised matrix FSCc . \ throughquad i s notWe an anymore invariant derive notions firsta local of property $ SL { defined2 } (R) by − $ invariants of cycles from theopentangents classic parenthesis in matrix the normal intersection invariants and focal closing point . parenthesis of cycles .. . orthogonalities Moreover comma , the .. focal see Section orthogonality .. 4 comma i s .. not both are defined in matrix t erms ofeven FSCc period symmetric . The corresponding notion of inversion ( in a cycle ) i s considered as MutualOrthogonalitywell disp . osition in generalised of two FSCc cycles i s not anymore may be a also local property characterised defined by through tangents in an the invariant notions of (intersection normalSection and point focal5 ..describ of cycles ) \ esquad period distancesorthogonalities .. Moreover and lengths comma defined .. , the\quad focal bysee orthogonality cycles Sect . ion Although i s\quad not .. even4 they , ..\quad symmetric shareboth some period are .. defined The in matrix t erms of FSCc . Orthogonalitycorrespondingstrange properties notion in generalisedof inversion ( e . g .open non FSCc parenthesis - lo i cal s character not in a anymore cycle or closing non a parenthesis -local symmetry property i s ) considered with defined the as orthogonalities well by period tangents in the intersectionSectionthey 5 are describ legiti point es distances - mate\quad objando f lengthsects c y c in l e defined Erlangens . \quad by cycles approachMoreover period since .. Although , \ theyquad are theythe conformal share focal some orthogonality under strange the M o¨ i s not \quad even \quad symmetric . \quad The correspondingpropertiesbius maps open parenthesis . notion We also of e period inversion consider g period the non ( in corresp hyphen a cycle lo onding cal character ) i perpendicularity s considered or non hyphen as symmetry and well it closing. s relation parenthesis t o with the orthogonalities they areorthogonality legiti hyphen . Invariance of “ infinitesimal ” cycles and corresp onding version of Sectionmateconformality obj 5 ects describ in Erlangen i s es considered distances approach in since Section and they lengths are 6 . conformal defined under by the cycles M o-dieresis . \quad bius mapsAlthough period .. they We also share some strange propertiesconsiderSection the corresp ( e . 7 onding g . deals non perpendicularity with− lo the cal global character .. and properties it s relation or non of t o the− .. orthogonalitysymmetry plane , e . ) periodg with . it .. s the Invarianceproper orthogonalities compact- .. of they are legiti − matequotedblleftification obj ects infinitesimal by in a zero Erlangen quotedblright - radius approach cycle .. cycles at since infinity and corresp they . Finally ondingare conformal , version Section of conformality 8 under considers the iMs some considered $ asp\ddot ects in{ Sectiono} of$ 6 bius period maps . \quad We a l s o considerSectionthe ..Cayley 7 the .. deals corresp transform with the onding global, with properties perpendicularity nicely interplays of the plane with comma\quad other eand period notions it g s period relation( e . it g s .proper focal t o compactification orthogonality\quad orthogonality by . \quad I n v a r i a n c e \quad o f ‘‘a zero infinitesimal, lengths hyphen , radius etc . cycle ’’ )\quad at considered infinitycycles period in and theFinally corresp previous comma onding sections Section 8 version . considers someof conformality asp ects of the Cayley i s considered transform comma in Section 6 . with nicely interplays with other notions open parenthesis e period g period focal orthogonality comma lengths comma etc period closing parenthesisSec tion \ ..quad considered7 \quad in deals with the global properties of the plane , e . g . it s proper compactification by athe zero previous− radius sections cycle period at infinity . Finally , Section 8 considers some asp ects of the Cayley transform , with nicely interplays with other notions ( e . g . focal orthogonality , lengths , etc . ) \quad considered in the previous sections . Erlangen Program at Large hyphen 1 : .... Geometry of Invariants .... 5 \noindenthlineErlangenErlangen Program Program at Large at - Large 1 :− 1 : \ h f i l l GeometryGeometry of of Invariants Invariants \ h f i l l 5 5 To finish this introduction we point out the following natural question period \ [ Problem\ r u l e {3em 1 period}{0.4 3 ptperiod}\ ] .... To which extend the subject presented here can be generalised to higher dimen hyphen sions ? 2 .. Elliptic commaTo finish .. parabolic this introduction .. and .. hyperbolic we point .. homogeneous out the following .. spaces natural question . \ centerlineWeProblem begin from{To representations 1 finish . 3 . thisTo of theintroduction which SL sub extend 2 open the we parenthesis subject point presented out R closing the parenthesis followinghere can be group natural generalised in Clifford question to algebras higher . with} two generators period Theydimen naturally - introduce circles comma parabolas and hyperbolas as invariant objects of corresp onding \noindentgeometriesProblem period 1 . 3 . \ h f i l l To which extend the subject presented here can be generalised to higher dimen − 2 period 1 SL sub 2 open parenthesis R closing parenthesissions ..? group and Clifford algebras \ beginWe consider{ a l i g n Clifford∗} algebras defined by elliptic comma parabolic and hyperbolic bilinear forms period .... Then s irepresentations o n2 s ? Elliptic of SL sub 2, open parenthesis parabolic R closing parenthesis and defined hyperbolic by the same formula homogeneous open parenthesis 2 period 3 closing parenthesis will\end inherit{spacesa l i g this n ∗} division period Convention 2 period 1 period .... There will be three different Clifford algebras C-lscript open parenthesis e closing parenthesis comma \noindentWe begin2 \quad from representationsE l l i p t i c , \quad of thep a rSL a b2 o(R l i) cgroup\quad inand Clifford\quad algebrash y p e r with b o l i c two\quad generatorshomogeneous . \quad spaces lscript-CThey open naturally parenthesis introduce p closing parenthesis circles , comma parabolas C-lscript and open hyperbolas parenthesis as h invariant closing parenthesis objects corresponding of corresp to ellipticonding comma geometries parabolic . comma and hyperbolic cases respectively period .... The notation C-lscript open parenthesis sigma closing parenthesis\noindent commaWe begin with assumed from representations values of the $ SL { 2 } ( R ) $ group in Clifford algebras with two generators . 2.1 SL2(R) group and Clifford algebras sigma = minus 1 comma 0 comma 1 comma .. refers to any of these three algebras period \noindentWe considerThey naturally Clifford algebras introduce defined circles by elliptic , parabolas , parabolic and and hyperbolas hyperbolic as bilinear invariant forms objects . of corresp onding A CliffordThen algebra C l open parenthesis sigma closing parenthesis as a 4 hyphen dimensional linear space i s spanned 1 by 1 comma e sub 0 commageometries e sub 1 comma . e sub 0 e sub 1 with representations of SL2(R) defined by the same formula ( 2 . 3 ) will inherit this division . non hyphen commutative multiplication defined by the following identities to the power of 2 : \noindentConvention$ 2 2 . . 1 1 . SL { 2 } ( R ) $There\quad willgroup be three and different Clifford Clifford algebras algebras Equation:C − lscript 2 e sub(e), 1 lscript = sigma− C =(p Case),C − 1lscript minus( 1h comma) corresponding for lscript-C open parenthesis e closing parenthesis endash elliptic case comma Case 2 0 comma for C-lscript open parenthesis p closing parenthesis endash parabolic case comma e sub 0 e sub 1 = minus e sub 1 e sub 0 period open \noindentto ellipticWe consider, parabolic Clifford, and hyperbolic algebras definedcases respectively by elliptic . , The parabolic notation andC − hyperboliclscript(σ), bilinear forms . \ h f i l l Then parenthesiswith 2 assumed period 1 closing values parenthesis Case 3 1 comma for lscript-C open parenthesis h closing parenthesis endash hyperbolic case comma .. 0 e toσ the= − power1, 0, 1, of 2refers = minus to 1any commaof these three algebras . \noindentThe two hyphenrepresentations dimensional subalgebra of $ SL of C-lscript{ 2 } open( parenthesis R ) $ e closing defined parenthesis by the .... same spanned formula by .... (1 .... 2 . and 3 i ) .... will = e sub inherit 1 e this division . A Clifford algebra C`(σ) as a 4 - dimensional linear space i s spanned 1 by 1, e0, e1, e0e1 sub 0 = minus e sub 0 e sub 1 .... i s .... is omorphic \noindentwith Convention 2 . 1 . \ h f i l l There will be three different Clifford algebras $ C−l s c r i p t ( e opennon parenthesis - commutative and can actuallymultiplication replace in definedall calculations by the ! closing following parenthesisidentities ....2 the: field of complex numbers C period .... For example comma) , l s c r i p t −C ( p ) , C−lscript ( h ) $ corresponding from open parenthesis 2 period 1 closing parenthesis follows that i to the power of 2 = open parenthesis e sub 1 e sub 0 closing parenthesis \noindent to elliptic , parabolic , and hyperbolic cases respectively . \ h f i l l The notation $ C−l s c r i p t to the power of 2 = minus 1 period .... For any C-lscript− 1, openforlscript parenthesis− C(e sigma) − −ellipticcase closing parenthesis, we identify R to the power of 2 with the ( \sigma )2 , $ with assumed values set of vectors 0e = −1, 2e1 = σ = 0, forC − lscript(p) − −paraboliccase, e0e1 = −e1e0. (2.1) w = ue sub 0 plus ve sub 1 comma where open parenthesis1, forlscript u comma− C(h) v− closing −hyperboliccase parenthesis, in R to the power of 2 period .. In the elliptic case\noindent of C-lscript$ open\sigma parenthesis= e− closing1 parenthesis , 0 this , maps 1 , $ \quad refers to any of these three algebras .

openThe parenthesis two - dimensional u comma v closing subalgebra parenthesis of C arrowright-mapsto− lscript(e) spanned e sub by0 open 1 parenthesisand i = ue1 pluse0 i= v closing−e0e parenthesis1 i s = e sub 0 z comma ..\ hspace withis .. omorphic∗{\z = uf i plus l l }A i v Clifford comma open algebra parenthesis $ 2 C period\ e 2 l closingl ( parenthesis\sigma ) $ as a 4 − dimensional linear space i s spanned 1 by $ 1 , e { 0 } , e { 1 } , e { 0 } e { 1 }$ with in the( and standard can actually form of complex replace numbers in all calculations period .. Similarly ! ) comma the seefield open of complex square bracket numbers 4 5 closingC. squareFor bracket and open square bracketexample 7 2 comma , Supplement C closing square bracket \noindent non − commutative multiplication2 2 defined by the following $ identities2 ˆ{ 2 } : $ openfrom parenthesis ( 2 . 1 p ) closing follows parenthesis that i = period (e1e0) ..= in− the1. For parabolic any caseC − lscript epsilon(σ =) ewe sub identify 1 e sub 0R openwith parenthesis the set such that epsilon to the power of 2 vectors = 0 closing parenthesis i s known as dual unit and all expressions \ begin { a l i g n ∗} u plusw = epsilonue + ve v comma, where u comma(u, v) ∈ v in2. R formIn the dual elliptic numbers case period of C − lscript(e) this maps 0 { e }ˆ{0 2 }1 = − 1R , \ tag ∗{$ 2 { e { 1 }} = \sigma = \ l e f t \{\ begin { a l i g n e d } & − 1 open parenthesis h closing parenthesis period .. in .. the(u, .. v) hyperbolic7→ e0(u+ i ..v case) = ..e0z, e = ewith sub 1 e subz = 0u+ openi v, parenthesis(2.2) such .. that .. e to the power, for ofin 2 the = 1 lscript standard closing− parenthesisC form ( of.. complex e is .. ) known numbers−− .. aselliptic .. double. Similarly .. unit case .. and, see .. , [ all 4\\ 5 ] and [ 7 2 , Supplement C &expressions] 0 , u plus f o rev comma C−l s cu r comma i p t v ( in R p constitute ) −− doubleparabolic numbers period case , e { 0 } e { 1 } = − e { 1 } e { 0 } . ( 2 . 1 ) \\ Remark( .. p 2 ) period . in 2 period the parabolic .. A part ..case of thisε = papere e ( cansuch b e that rewrittenε2 = in 0) ti erms s known of complex as dual comma unit .. dualand and all double & 1 , for lscript −C ( h1 ) 0 −− hyperbolic case , \end{ a l i g n e d }\ right . $ } numbersexpressions and it willu + haveεv, u, some v ∈ R commonform dual points numbers with Supplement. C of the book open square bracket 7 2 closing square bracket period .. \end{ a l i g n ∗} 2 However ( h ) . in the hyperbolic case e = e1e0 ( such that e = 1) is theknown usage of Clifford as algebrasdouble provides unit some facilitiesand which all expressionsdo not have naturalu + ev, equivalent u, v ∈ R inconstitute double \noindentcomplexnumbers numbersThe. two comma− dimensional see Remark 4 period subalgebra 1 3 period of .. Moreover $ C−lscript the language ( of e Clifford )$ algebras\ h f i isl l morespanned uniform by \ h f i l l 1 \ h f i l l and i \ h f i l l $ =andRemark also e { allows1 } straightforwarde 2 .{ 20 .} generalisations= A− parte { t of o0 higher this} e paper dimensions{ 1 can}$ open b\ eh rewritten square f i l l i bracket s in\ h t 3 f ierms 5 l closingl i s of omorphic squarecomplex bracket , period hlinedual and double numbers and it will have some common points with Supplement C of the \noindent1 subbook We label[( 7 and 2 generators ] . can However actually of our Clifford the replace usage algebra of in byClifford all e sub calculations 0 algebras and e sub provides 1 following ! ) \ someh the f i l C lfacilities slashthe C field plus which plus of do indexing complex not agreement numbers which $ isC . $ \ hused f i lhave l byFor computer natural example algebra equivalent , calculations in complex in open numbers square bracket , see 4 Remark 6 closing 4 square . 1 3 bracket . Moreover period the language 2 subof In Clifford light of algebras usefulness is of more infinitesimal uniform numbers and also open allows square bracketstraightforward 1 8 comma generalisations 6 9 closing square t bracket o higher in the parabolic spaces open parenthesis\noindentdimensions seefrom Section[ ( 3 26 5 period . ] . 1 ) 1 closingfollows parenthesis that $ it may i ˆ{ be2 } = ( e { 1 } e { 0 } ) ˆ{ 2 } = − 1 . $ \ hworth f i l l For to consider any the $ C parabolic−l s c r i p Cliffordt ( algebra\sigma C-lscript) open $ we parenthesis identify epsilon $Rˆ closing{ 2 parenthesis}$ with with the a generator set of 2 vectors e sub 1 = epsilon comma where epsilon is an infinitesimal number period \noindent $ w = ue { 0 } + ve { 1 } ,$ where $( u , v ) \ in R ˆ{ 2 } . $ \quad In the elliptic case of $ C−lscript ( e )$ thismaps 1We label generators of our Clifford algebra by e0 and e1 following the C/C ++ indexing agreement which is \ hspaceused∗{\ byf computer i l l } $ algebra ( u calculations , v in [ 4) 6 ] .\mapsto e { 0 } ( u + $ i $ v ) = e { 0 } z , $ \quad with2In light\quad of usefulness$z=u+$i$v of infinitesimal numbers [ 1 8 , 6 9 , ] in the ( parabolic 2 . spaces 2 ( see )$ Section 6 . 1 ) it may be

worth to consider the parabolic Clifford algebra C − lscript(ε) with a generator 2e1 = ε, where ε is an infinitesimal \noindentnumberin . the standard form of complex numbers . \quad Similarly , see [ 4 5 ] and [ 7 2 , Supplement C ]

( p ) . \quad in the parabolic case $ \ varepsilon = e { 1 } e { 0 } ( $ such that $ \ varepsilon ˆ{ 2 } = 0 ) $ i s known as dual unit and all expressions $ u + \ varepsilon v , u , v \ in R $ form dual numbers .

( h ) . \quad in \quad the \quad h y p e r b o l i c \quad case \quad $ e = e { 1 } e { 0 } ( $ such \quad that \quad $ e ˆ{ 2 } = 1 ) $ \quad i s \quad known \quad as \quad double \quad unit \quad and \quad a l l expressions $u + ev , u , v \ in R $ constitute double numbers .

\noindent Remark \quad 2 . 2 . \quad A part \quad of this paper can b e rewritten in t erms of complex , \quad dual and double numbers and it will have some common points with Supplement C of the book [ 7 2 ] . \quad However the usage of Clifford algebras provides some facilities which do not have natural equivalent in complex numbers , see Remark 4 . 1 3 . \quad Moreover the language of Clifford algebras is more uniform and also allows straightforward generalisations t o higher dimensions [ 3 5 ] .

\ begin { a l i g n ∗} \ r u l e {3em}{0.4 pt } \end{ a l i g n ∗}

\ hspace ∗{\ f i l l } $ 1 { We }$ label generators of our Clifford algebra by $ e { 0 }$ and $ e { 1 }$ following the C / C $ + + $ indexing agreement which is

\noindent used by computer algebra calculations in [ 4 6 ] .

$ 2 { In }$ light of usefulness of infinitesimal numbers [ 1 8 , 6 9 ] in the parabolic spaces ( see Section 6 . 1 ) it may be worth to consider the parabolic Clifford algebra $ C−l s c r i p t ( \ varepsilon ) $ with a generator $ 2 { e { 1 }} = \ varepsilon , $ where $ \ varepsilon $ is an infinitesimal number . 6 .... V period V period Kisil \noindenthline6 6 \ h f i l l V . V . K i s i l V . V . Kisil We denote the space R to the power of 2 of vectors ue sub 0 plus ve sub 1 by R to the power of e comma R to the power of p or R to the power\ [ \ r of u l h e { t3em o highlight}{0.4 pt which}\ ] of Clifford

algebras i s used in the present context2 period .. The notation Re top the powerh of sigma assumes C-lscript open parenthesis sigma closing We denote the space R of vectors ue0 + ve1 by R , R or R t o highlight which of Clifford parenthesis period σ WeThe denotealgebras SL sub the 2 i open s space used parenthesis in $ the R ˆ present{ R2 closing}$ context parenthesis of vectors . group The $ notation open ue square{ 0 R} bracketassumes+ 2ve 7 commaC{−1lscript} ..$ 5(σ by 5). comma $ R .. ˆ{ 6e 8 closing} , square R ˆ{ bracketp }$ consistsor $ R of 2ˆ{ timesh } 2$ matricesThe t oSL highlight2(R) group which [ 2 7 , of 5Clifford 5 , 6 8 ] consists of 2 × 2 matrices algebras i s used a in b the present context . \quad The notation $ R ˆ{\sigma }$ assumes $ C−l s c r i p t Row 1 a b Row 2 c d . comma, ..with .. aa, comma b, c, d ∈ bR commaand c commathe determinant d in R .. and thead determinant− bc = 1. .. ad minus bc = 1 period ( An\sigma isomorphic) realisation .c $ d of SL sub 2 open parenthesis R closing parenthesis .... with the .... same multiplication i s .... obtained if we replaceAn isomorphic realisation of SL2(R) with the same multiplication i s obtained if we replace \ centerline {The a $ b SL { 2 } a( beR )$ group[27, \quad 5 5 , \quad 6 8 ] consists of $ 2 \times a matrixa matrix .... Row 1 a b Rowby 2 c d . .... by ....0 Rowwithin 1 a be any sub 0C Row− lscript 2 minus(σ). ceThe sub 0 advantage d . .... within of theany C-lscriptlatter open parenthesis sigma closing2 $ matrices parenthesis} periodc d .... The advantage−ce0 ofd the latter form i s that weform can define i s that the M o-dieresis bius transformation of R to the power of sigma right arrow R to the power of sigma for all three algebras σ σ C-lscript\ centerlinewe open can parenthesis{ define$\ l e fthe t ( sigma\Mbegin closingo¨ bius{ array parenthesis transformation}{ cc } a by the& same bof \\R →cR &for d all\endthree{ array algebras}\ rightC)− lscript , $ (\σquad) with \quad $ a ,Line bby 1 expression the , same c : , Line d 2 Row\ in 1 a beR sub $ 0 Row\quad 2 minusand ce the sub determinant 0 d . : ue sub 0\ plusquad ve sub$ ad1 arrowright-mapsto− bc = a open 1 parenthesis . $ } ue sub 0 plus ve sub 1 closing parenthesis plus be sub 0 divided by minus ce sub 0 open parenthesis ue sub 0 plus ve sub 1 closing parenthesis plus d expression : sub\noindent comma openAn parenthesis isomorphic 2 period realisation 3 closing parenthesis of $ SL { 2 } ( R ) $ \ h f i l l with the \ h f i l l same multiplication i s \ h f i l l obtained if we replace   where the expression a divideda by be b0 in a non hyphen commutativea(ue algebra0 + ve is1)always + be0 understood as ab to the power of minus 1 comma see \noindent a matrix \ h f i l l $\ l e f t (\: beginue0 +{vearray1 7→}{ cc } a & b \\ c(2 &.3) d \end{ array }\ right )$ \ h f i l l by \ h f i l l open square bracket 1 .. 5 comma−ce0 1 ..d 6 closing square bracket period−ce0(ue0 + ve1) + d , $\ lTherefore e f t (\ begin ac divided{ array by}{ bccc =} aa divided & be by b{ but0 ca}\\ divided − byce cb equal-negationslash{ 0 } & d \end a{ dividedarray }\ byright b in general)$ \ periodh f i l l within any $ C−l s c r i p t ( \sigma ) . $ \ ha f i l l The advantage of the latter form i s that −1 Againwhere in the the elliptic expression case theb transformationin a non - commutative .. open parenthesis algebra 2 period is always 3 closing understood parenthesis as .. iab s equivalent, see [ 1 t o comma .. cf period .. ac a ca a open square5 , 1 bracket 6 ] . 6 commaTherefore .. Chapterbc = b 1but 3 closingcb 6= squareb in general bracket comma . .. open square bracket 5 comma \noindentChapterAgain 3we closing canin the square define elliptic bracket the case : M Row the $ 1 transformation\ addot be sub{o} 0$ Row bius 2 minus ( 2 transformation . ce3 ) sub 0i sd equivalent . : e sub of 0 $z t mapsto-arrowright Ro , ˆ{\ cfsigma . [ 6}\ ,e subrightarrow 0 open parenthesisR ˆ{\ a sigma }$ openfor parenthesis allChapter three 1u algebrasplus3 ] , e sub [ 5 1 , e $ sub C− 0l s v c closing r i p t parenthesis ( \sigma plus b closing) $ parenthesis by the same divided by c open parenthesis u plus e sub 1 e sub 0 v closing parenthesis plus d = e sub 0 az plus b divided by cz plus d sub comma where z = u plus i v comma \ [ \whichbegin i{ sa the l i g standard n e d } expression form of a M o-dieresis : \\ bius transformation period .. One can straightforwardly verify that \ lthe e f t map(\ begin open parenthesis{ array }{ cc 2 period} a &3 closing be parenthesis{ 0 }\\ i − s a leftce action{ 0 of} SL& sub d 2 open\end parenthesis{ arrayChapter3]}\ right R : closing) parenthesis : ue { on0 R to} the+ power ve { 1 } \mapsto \ f r a c { a ( ue { 0 } + ve { 1 } ) + be { 0 }}{ − ce { 0 } ( ue { 0 } + ve { 1 } of sigma comma i perioda e period be0 g 1 open parenthesise0(a g(u 2+ toe the1e0v power) + b) of w closingaz + b parenthesis = open parenthesis g 1 g 2 closing parenthesis ) + d } { , } ( 2: .e0z 37→ ) \end{ a l i g n e d }\ ] = e0 where z = u + iv, w period −ce0 d c(u + e1e0v) + d cz + d , To study finer structure of M o-dieresis bius transformations it i s useful t o decompose an element g of SLwhich sub 2 i open s the parenthesis standard R form closing of a parenthesis M o¨ bius into transformation the product g .= g a One g n gcan k to straightforwardly the power of : verify \noindent where the expression $\ f r a c { a }{ b }$ inσ a non − commutativew algebra is always understood as Equation:that the open map parenthesis ( 2 . 3 2 ) period i s a left 4 closing action parenthesis of SL2(R ..Row) on 1R a, bei . sub e .g 01( Rowg2 ) 2 = minus (g1g2) cew. sub 0 d . = alpha from 0 to parenleftbigg minus$ ab to ˆ{ theTo − power study1 } of finer 1, alpha $ structure see to the [ power 1 of\ Mquad ofo¨ 0bius parenrightbigg5 , transformations 1 \quad Row6 ]1 1 . itnu i e s sub useful 0 Row t o 2 decompose 0 1 . Row 1 cosine an element phi e sub 0 sine phi Row 2 e sub Therefore $\ f r a c { ac }{ bc } = \ f r a: c { a }{ b }$ but $\ f r a c { ca }{ cb }\ne \ f r a c { a }{ b }$ in general . 0 sine phig of cosineSL2( phiR) into . comma the product g = gagngk where the values of parameters are as follows : \ hspace ∗{\ f i l l }Again in the elliptic case the transformation \quad ( 2 . 3 ) \quad i s equivalent t o , \quad c f . \quad [ 6 , \quad Chapter 1 3 ] , \quad [ 5 , Equation: open parenthesis 2 period 5 closing parenthesis .. alpha =  radicalbig-line of c to the power of 2 plus d to the power of 2 sub comma nu = ac plus bd commaa phi =be0 minus arctan0 c1 divided0 by1 dνe sub0 periodcos φ e0 sin φ = α( − α ) , (2.4) \ beginConsequently{ a l i g n ∗} cos phi = d− dividedce0 d by square root of c to the0 power 1 of 2 pluse0 sin dφ to thecos powerφ of 2 and sin phi = minus c divided by square Chapter 3 ] : \\\ l e f t (\ begin { array }{ cc } a & be { 0 }\\ − ce { 0 } & d \end{ array }\ right ) root ofwhere c to the the power values of 2 of plus parameters d to the power are of as 2 follows period .... : The product open parenthesis 2 period 4 closing parenthesis gives a realisation of: the e { 0 } z \mapsto \ f r a c { e { 0 } ( a ( u + e { 1 } e { 0 } v ) + b ) }{ c (Iwasawa u + .. decomposition e { 1 } e .. open{ 0 square} v bracket ) + 5 5 comma d } = .. S ..e III{ period0 }\ 1f r closing a c { az square + bracket b }{ ..cz in the + form d SL} sub{ , 2} openwhere z = u + i v , parenthesis R closing parenthesis = ANK commap 2 ..2 where K is the maximal c \end{ a l i g n ∗} α = c + d , ν = ac + bd, φ = − arctan (2.5) compact group comma N i s nilp otent and A normalises N period d . 2 period 2 .. Actions of subgroups \noindentConsequentlywhich i cos s theφ = standard√ d and form sin φ of= √ aM−c . $ \Theddot product{o} $ (bius 2 . 4 transformation ) gives a realisation . \quad of One can straightforwardly verify that We describ e here orbits of the ..c2+ threed2 subgroups fromc2+ thed2 Iwasawa decomposition open parenthesis 2 period 4 closing parenthesis .. for all thethree mapthe typ ( es 2 of . Clifford 3 ) i algebras s a left period action However of there $SL are less{ than2 } nine( open R parenthesis ) $ on = $ 3 Rtimes ˆ{\ 3 closingsigma parenthesis} , $ idifferent . e orbits $ . g 1 ( g 2 ˆ{ w } )=(g1g2)w.$ since Iwasawa decomposition [ 5 5 , § III . 1 ] in the form SL2(R) = ANK, where in allK threeis the EPH maximal cases the compact subgroups group A and,N N acti s through nilp otent M dieresis-o and A biusnormalises transformationN. uniformly : ToLemma study2 . 2 finer period Actions 3 structure period of.. For subgroups of any M type $ \ ofddot the Clifford{o} $ algebra bius C-lscript transformations open parenthesis it isigma s useful closing parenthesis t o decompose : an element $ g1 period $We describ .. The subgroup e here orbits N defines of shi the f-t sthree ue sub 0subgroups plus ve sub from 1 mapsto-arrowright the Iwasawa open decomposition parenthesis u ( plus 2 . nu 4 closing parenthesis e sub 0o plus f ve)$ SLsub for 1{ .. all2 along}three the( ..typ R quotedblleft es )$ of Clifford intotheproduct real quotedblright algebras ... axisHowever $g U by nu there= period g are aless gthan n nine g(= kˆ 3 ×{ 3): }$ Thedifferent vector field orbits of the since derived in representation all three EPH is dN cases sub athe open subgroups parenthesisA uand commaN vact closing through parenthesis M o¨ =bius open parenthesis 1 comma 0 closing\ begintransformation parenthesis{ a l i g n ∗} period uniformly : \ l e2 f periodt Lemma(\ begin .. The{ 2array subgroup . 3}{ . cc A} For definesa &any dilations be type{ of ue0 the sub}\\ Clifford 0 plus − vece algebra sub{ 1 mapsto-arrowright0 }C &− lscript d \end(σ):{ alphaarray to}\ theright power) of = minus\alpha 2 openˆ parenthesis{ 0 } { ( ue − }ˆ{ 1 }\alpha ˆ{ 0 } ) \ l e f t (\ begin { array }{ cc } 1 & \nu e { 0 }\\ 0 & 1 \end{ array }\ right ) \ l e f t (\ begin { array }{ cc }\cos sub 0 plus1 ve . sub 1 The closing subgroup parenthesisN ..defines by the factorshi f alpha− t s toue the0 + powerve1 7→ of( minusu + ν)e 20 which+ ve1 fixesalong the “ real \phi & e { 0 }\ sin \phi \\ e { 0 }\ sin \phi & \cos \phi \end{ array }\ right ), \ tag ∗{$ ( origin” open axis parenthesisU by 0ν. commaThe vector 0 closing field parenthesis of the derived period .. representation The vector field of is thedN deriveda(u, v) representation = (1, 0). is dA sub a open parenthesis 2 . 4 ) $} −2 −2 u comma v2 closing . The parenthesis subgroup = openA defines parenthesis dilations 2 u commaue0 + 2ve v closing1 7→ α parenthesis(ue0 +ve1) periodby the factor α which \end{ a l i g n ∗} fixes origin ( 0 , 0 ) . The vector field of the derived representation is dAa(u, v) = (2u, 2v). \noindent where the values of parameters are as follows :

\ begin { a l i g n ∗} \alpha = \ sqrt { c ˆ{ 2 } + d ˆ{ 2 }} { , }\nu = ac + bd , \phi = − \arctan \ f r a c { c }{ d } { . }\ tag ∗{$ ( 2 . 5 ) $} \end{ a l i g n ∗}

\noindent Consequently cos $ \phi = \ f r a c { d }{\ sqrt { c ˆ{ 2 } + d ˆ{ 2 }}}$ and s i n $ \phi = \ f r a c { − c }{\ sqrt { c ˆ{ 2 } + d ˆ{ 2 }}} . $ \ h f i l l The product ( 2 . 4 ) gives a realisation of the

\noindent Iwasawa \quad decomposition \quad [ 5 5 , \quad \S \quad I I I . 1 ] \quad in the form $ SL { 2 } ( R ) = ANK , $ \quad where $ K $ is the maximal compact group $ , N$ i s nilp otent and $A$ normalises $N . $

\noindent 2 . 2 \quad Actions of subgroups

\noindent We describ e here orbits of the \quad three subgroups from the Iwasawa decomposition ( 2 . 4 ) \quad f o r a l l three typ es of Clifford algebras . However there are less than nine $ ( = 3 \times 3 ) $ different orbits since in all three EPH cases the subgroups $ A $ and $ N $ act throughM $ \ddot{o} $ bius transformation uniformly :

\noindent Lemma 2 . 3 . \quad For any type of the Clifford algebra $ C−l s c r i p t ( \sigma ) : $

1 . \quad The subgroup $ N $ defines shi $ f−t $ s $ ue { 0 } + ve { 1 }\mapsto ( u + \nu ) e { 0 } + ve { 1 }$ \quad along the \quad ‘ ‘ r e a l ’ ’ \quad a x i s $ U $ by $ \nu . $ The vector field of the derived representation is $ dN { a } ( u , v ) = ( 1 , 0 ) . $

2 . \quad The subgroup $ A $ defines dilations $ ue { 0 } + ve { 1 }\mapsto \alpha ˆ{ − 2 } ( ue { 0 } + ve { 1 } ) $ \quad by the factor $ \alpha ˆ{ − 2 }$ which fixes origin ( 0 , 0 ) . \quad The vector field of the derived representation is $ dA { a } ( u , v )=(2u,2v).$ Erlangen Program at Large hyphen 1 : .... Geometry of Invariants .... 7 \noindenthlineErlangenErlangen Program Program at Large at - Large 1 :− 1 : \ h f i l l GeometryGeometry of of Invariants Invariants \ h f i l l 7 7 Figure 1 period .. Actions of the subgroups A and N by M o-dieresis bius transformations period \ [ Orbits\ r u l e and{3em vector}{0.4 fields pt }\ corresponding] t o the derived representation .. open square bracket 3 0 comma .. S 6 period 3 closing square bracket comma .. open square bracket 5 5 comma Chap hyphen t er VI closing squareFigure bracket 1 . of theActions Lie algebra of the subgroups s l sub 2 forA and subgroupsN by M Ao ¨andbius N transformations are shown in Fig . period 1 period .. Thin transverse lines \ centerlinej oin pointsOrbits{ ofFigure orbits and vector corresponding 1 . \ fieldsquad corresponding tActions o the same of values the t o of thesubgroups thederived parameter representation $A$ along the and subgroup $N$[ period 3 0 byM , §6 $ . 3\ddot ] , {o} $ bius transformations . } By[ .. 5 contrast 5 , Chap .. the - .. actions .. of the .. subgroup .. K .. look .. differently between the EPH .. cases comma .. see \ hspaceFigt period er∗{\ VIf 2 i period ]l l of} Orbits the .. They Lie and obviously algebra vector correlatesl2 fieldsfor subgroupswith corresponding names chosenA and for C-lscriptN t oare the shown open derived parenthesis in Fig representation . e 1closing . parenthesis Thin\quad comma[ 3 lscript-C 0 , \quad open \S 6 . 3 ] , \quad [ 5 5 , Chap − parenthesistransverse p closing lines parenthesis j oin points comma of C-lscript orbits corresponding open parenthesis t h o closing the same parenthesis values period of the .. parameter However algebraic along \noindentexpressionsthe subgroupt for er these VI . orbits ] of are the uniform Lie algebra period $ s l { 2 }$ for subgroups $A$ and $N$ are shown in Fig . 1 . \quad Thin transverse lines jclosing oin pointsBy parenthesis contrast of = orbits from closing the corresponding parenthesis actions = t toof o Vector the the sub same subgroup dK values sub p openK of parenthesis thelook parameter differentlyu from dK along sub between h the open subgroup parenthesis u . to dK sub e open parenthesisthe EPH u sub cases comma , sub see comma Fig v sub . 2v to . the power They of obviouslyfields comma correlate v closing parenthesis with names to the chosen power forof = are sub open parenthesis By1 from\quadC open− lscriptc parenthesis o n t r(e a) s, tlscript\ 1quad to− openCthe(p) parenthesis,C\quad− lscripta 1 c( th plus i) o. n s fromHowever\quad plus too algebraic f plus the :\ uquad sub expressions commasubgroup from for u\quad tothese the powerorbits$ K $ of are 2\quad to u tolook the power\quad of 2differently to between the EPH \quad c a s e s , \quad see theFig poweruniform . 2 of . 2\ subquad . plusThey to the obviously power of minus correlate comma from with v to names the power chosen of 2 comma for to $ Cv− tolscript the power of ( 2 2 uve from ) 2 uv , to lscript2 uv closing−C ( p ) , C−lscript ( h ) .$ \quad However algebraic parenthesis from closing parenthesis to)= closing parenthesis 2 − v2, 2uv ) expressions for these orbits) = arefields uniformare . (1 + : u u2 +, 2 2uv2uv ) Figure 2 period .. Action of the K subgroup,v period) = The corresponding(1 + , orbitsv are circles) comma parabolas and hyperbolas period Vector (1 + u2 dKh(u ,,vv dKp(u Lemma 2 period 4 period .. A K hyphen orbitdK ine(u R to the power of sigma passing the point open parenthesis 0 comma t closing parenthesis has\ [ the ) following = ˆ{ ) equation = } :{ Vector { dK { p } ( u ˆ{ dK { h } ( u } { dK { e } ( u }}ˆ{ f i e l d s { , v )Figure}} { 2, . { Action, } v of the{ vK}}subgroupˆ{ = }} . Theare corresponding{ ( 1 orbits ˆ{ ( are circles 1 } { , parabolas( 1 } and+ hyperbolas ˆ{ + } { . + }} : u ˆ{ 2 } { , ˆ{ u ˆ{ 2 }} { u ˆ{ 2 }}}ˆ{ − } { + } open parenthesis u to the power of 2 minus sigmaσ v to the power of 2 closing parenthesis minus 2 v t to the power of minus 1 minus sigma t , ˆ{ Lemmav ˆ{ 2 } 2 ., 4} . { vA ˆ{ K2 −}}orbit2 in uvR ˆ{passing2 uv the} point{ 2(0 uv, t) }has) the ˆ{ following) } { ) equation}\ ] : divided by 22 plus 12 = 0 commat−1−σt .. where sigma = 2 e sub 1 open parenthesis i period e period minus 1 comma 0 or 1 closing parenthesis period (u − σv ) − 2v 2 + 1 = 0, where σ = 2e1 ( i . e . − 1, 0 or 1 ) . ( 2 . 6 ) The .. opencurvature parenthesis of 2 period a K− 6 closingorbit at parenthesis point (0, t) is equal to The curvature of a K hyphen orbit at point open parenthesis 0 comma t closing parenthesis .. is equal to \ centerline { Figure 2 . \quad Action of the $ K $ subgroup . The corresponding orbits are circles , parabolas and hyperbolas . } Line 1 2 t Line 2 kappa = overbar 1 sub plus sigma t 2 period2t A proof will be given lat er open parenthesis see Example 3 period 4 period 2 closing parenthesis comma .. when a more suitable t ool will κ = 1 2. be\noindent in our Lemma 2 . 4 . \quad A $ K − $ orbit+σt in $Rˆ{\sigma }$ passing the point $( 0 , tdisposal ) $A period has proof the .. will Meanwhile following be given these lat equation formulae er ( see allows Example : t o produce 3 . 4 . geometric 2 ) , when characterisation a more suitable of K hyphen t ool orbits will beperiod Lemmain our 2 period disposal 5 period . Meanwhile these formulae allows t o produce geometric characterisation $ ( u ˆ{ 2 } − \sigma v ˆ{ 2 } ) − 2 v \ f r a c { t ˆ{ − 1 } − \sigma t }{ 2 } + 1 openof parenthesisK− orbits e closing. parenthesis period .. For C-lscript open parenthesis e closing parenthesis the orbits of K are circles comma they =are coaxal 0Lemma open , $ square 2\quad . 5 bracket .where 1 7 comma $ \sigma S 2 period= 3 closing2 { e square{ 1 bracket}} ( .. $ with i the .real e $line . being− the1 , 0 $ or 1 ) . \quad ( 2 . 6 ) The curvature of a $ K − $ orbitatpoint $( 0 , t )$ \quad i s equal to radical axis( e period ) . ..For A circleC with− lscript centre(e) atthe open orbits parenthesis of K 0are comma circles open , parenthesis they arecoaxal v plus v to[ 1 the 7 ,power§2 . of 3 minus] 1 closing parenthesis slash 2with closing the parenthesis real line passing being the through two points open parenthesis 0 comma v closing parenthesis .. and \ [ \ begin { a l i g n e d } 2 t \\ open parenthesisradical axis 0 comma . v Ato thecircle power with of centreminus 1 at closing(0, parenthesis(v + v−1)/2) periodpassing .. The through vector two field points of the derived(0, v) representation is dK sub e open\kappa parenthesisand = u\ commaoverline v closing{\}{ parenthesis1 } { + = open\sigma parenthesist } u to2 the power . \end of{ 2a minus l i g n e dv}\ to] the power of 2 plus 1 comma 2 uv closing −1 2 2 parenthesis(0 period, v ). The vector field of the derived representation is dKe(u, v) = (u − v + 1, 2uv). A proof will be given lat er ( see Example 3 . 4 . 2 ) , \quad when a more suitable t ool will be in our d i s p o s a l . \quad Meanwhile these formulae allows t o produce geometric characterisation of $ K − $ o r b i t s .

\noindent Lemma 2 . 5 .

\ hspace ∗{\ f i l l }( e ) . \quad For $ C−lscript ( e ) $ the orbits of $K$ are circles , they are coaxal [ 17 , \S 2 . 3 ] \quad with the real line being the

\ hspace ∗{\ f i l l } radical axis . \quad Acirclewithcentreat $( 0 , ( v + vˆ{ − 1 } ) / 2 )$ passingthroughtwopoints $( 0 , v )$ \quad and

\ centerline { $ ( 0 , v ˆ{ − 1 } ) . $ \quad The vector field of the derived representation is $ dK { e } ( u , v ) = ( u ˆ{ 2 } − v ˆ{ 2 } + 1 , 2 uv ) . $ } 8 .... V period V period Kisil \noindentLine8 1 hline8 Line\ h f 2i l open l V parenthesis . V . K i as i closing l parenthesis open parenthesis b closing parenthesis V . V . Kisil Figure 3 period K hyphen orbits as conic sections : .... open parenthesis a closing parenthesis a flat projection along U axis semicolon open parenthesis\ [ \ begin { ba closingl i g n e d parenthesis}\ r u l e {3em same}{ values0.4 pt of}\\ phi on different ( a ) ( b ) \end{ a l i g n e d }\ ] orbits belong to the same generator of the cone period(a) (b) open parenthesis p closing parenthesis period .. For C-lscript open parenthesis p closing parenthesis the orbits of K are parabolas with the verticalFigure axis V period3.K− ..orbits A parabola as conic passing sections through : ( a ) a flat projection along U axis ; ( b ) same values of φ on \noindentopendifferent parenthesisFigure 0 comma $ 3 v slash . 2 K closing− $ parenthesis orbits has as horizontal conic sections directrix passing : \ h through f i l l ( open a ) parenthesis a flat projection 0 comma v minus along v to the $U$ poweraxis oforbits ; minus ( b belong ) 1 slashsame to the2 values closing same parenthesisgenerator of $ \ ofphi and the focus$ cone on .at opendifferent parenthesis 0 comma open parenthesis v plus v to the power of minus 1 closing parenthesis( slash p ) . 2 closing For parenthesisC − lscript period(p) the orbits of K are parabolas with the vertical axis V. A −1 \noindentTheparabola vectororbits field passing of the belong derived through to representation the(0, v/ same2) has is generator dK horizontal sub p open of directrix parenthesis the cone passing u comma. through v closing parenthesis(0, v − v / =2) openand parenthesis u to the power −1 2 of 2 plusfocus 1 comma at (0 2, uv(v + closingv )/2) parenthesis. The vector period field of the derived representation is dKp(u, v) = (u +1, 2uv). ( popen ) . parenthesis(\quad h ) .For h closing For $ C−C parenthesislscript− lscript( periodh) (the .. pFor orbits C-lscript ) $ of theK openare orbits parenthesis hyperbolas of h $K$ closingwith asymptotes parenthesis are parabolas .. parallel the orbits with to of the K are vertical hyperbolas axis with $ Vlines . $ u\quad= ±v.AA parabola hy - passing through asymptotes parallel to lines u = plusminux v period .. A hy hyphen 2 $( 0perbola , v passing / 2 through )$ the hashorizontal point (0, v) has directrix the focal passingthrough distance 2p, where $(p = 0v√+1 , v − v ˆ{ − perbola passing through the point open parenthesis 0 comma v closing parenthesis .. has the focal distance 22 pv comma .. where p = v to the1 } powerand/ of the 2 2 plus )$andfocusat 1 divided by square root $( of 2 v .. 0 and , the ( v + vˆ{ − 1 } ) / 2 ) . $ Theupper vector focus is field located of at openthe parenthesisderivedupper focus representation 0 comma is located f closing at parenthesis is(0, f $) dKwith .. with{ :p :} ( u , v ) = ( u ˆ{ 2 } + 1f = , braceleftmid-braceleftbt 2 uv ) . $ p to the power of p minus plus radicalBig-line of p 2 divided by 2 minus 1 radicalBig-line of p 2 divided by 2 r r , minus 1 sub comma to the power of comma for to thep power of2 for 0 v less2 greater equalfor v 1 sub period less 1 comma \ hspace ∗{\ f i l l f}(= hbraceleftmid ) . \quad− braceleftbtFor $ C−lscript p − + p − (1 p h− 1 )$for\quad0v the<≥ v orbits1. < 1, of $ K $ are hyperbolas with asymptotes parallel to lines The vector field of the derived representation is dK sub h open2 parenthesis2 u, comma v closing parenthesis = open parenthesis u to the power of$ 2u plus = v to\ thepm powerv of 2 . plus $ 1\quad commaA 2 hy uv closing− parenthesis period dK (u, v) = (u2 + v2 + 1, 2uv). Since all K hyphenThe vector orbits are field conic of thesections derived it is t representation empting t o obtain is themh as sections of some cones period K− \ hspaceTo this∗{\ endSincef iwe l l define all} perbola theorbits family passing are of double conic through hyphen sections the sided it point is right t empting hyphen $ ( angle t 0o obtain cones , b themv e parametrised ) as $ sections\quad by t of greaterhas some the 0 : focal distance $ 2 pEquation: ,cones $ \quad .open parenthesiswhere $ 2 p period = 7 closing\ f r a c { parenthesisv ˆ{ 2 ..} x+ to the 1 power}{\ sqrt of 2 plus{ 2 parenleftbig} v }$ y\quad minus 1and divided the by 2 parenleftbig t plus t > 0 : t to theTo power this of end minus we 1 define parenrightbig the family parenrightbig of double to - the sided power right of 2 - minus angle parenleftbig cones b e parametrised z minus 1 divided by by 2 parenleftbig t minus t to the power\ centerline of minus{ 1upperfocus parenrightbig parenrightbig is located to at the power $( of 0 2 = 0 , period f )$ \quad with : } The vertices of cones b elong t o the hyperbola1 open brace x = 01 comma y to the power of 2 minus z to the power of 2 = 1 closing brace \ [ f = braceleftmid −braceleftbtx2 + (y − (t p+ t ˆ−{1))p2 −}(z −{− +(t}\− t−sqrt1))2 {= 0p. \ f r a c { 2 }{ 2 } −(2.7)1 }\ sqrt { p \ f r a c { 2 }{ 2 } comma see Fig period .. 3 for illustration period2 2 − Lemma1 }ˆ{ 2 period, } { 6, period} f K o r hyphen ˆ{ f o orbits r } may0 be{ obtainedv } < cases{\ asgeq fo l} lowsv{ :1 } { . } < 1 , \ ] 2 2 openThe parenthesis vertices e of closing cones parenthesis b elong t period o the .. hyperbola e l liptic K hyphen{x = 0, orbits y − z are= s 1 ections}, see Fig of cones . open 3 for parenthesis illustration 2 period 7 closing parenthesis .. by the. plane z = 0 open parenthesis EE to the power of prime on Fig period .. 3 closing parenthesis semicolon \ centerlineopenLemma parenthesis{The2. p6 vector.K closing− orbits parenthesis field may of period be the obtained .. derived parabolic cases representation K hyphenas fo l orbits lows :are is s ections $ dK of open{ h parenthesis} ( u 2 period , 7 v closing ) parenthesis = ( .. u ˆ{ 2 } 0 ( e+by ) the . v p ˆ lane{ e l2 lipticy} = plusminux+K− 1orbits z , open are 2 parenthesis s uvections ) PP of to cones . the $ power}( 2 .of 7 prime ) onby Fig the period plane 3 closingz = 0(EE parenthesison Fig semicolon . 3 ) ; 0 ( popen ) . parenthesis parabolic h closingK− parenthesisorbits are period s ections .. hyperbolic of ( 2 K. 7 hyphen ) by orbits the are p lane s ectionsy = of± openz(PP parenthesison Fig . 2 period3 ) ; 7 closing parenthesis 0 (..\ hhspace by ) . the p∗{\ lane hyperbolicf i y l l =} 0Since openK− parenthesis aorbits l l $ areK HH s− to ections$ the powerorbits of of( prime2 are . 7 conic )on Figby period sections the p .. lane 3 closing ity is= parenthesis 0( tHH emptingon periodFig t .o obtain 3 ) . them as sections of some cones . Moreover commaMoreover each straight , each line straight generating line a cone generating from the family a cone open from parenthesis the family 2 period( 2 7 . closing 7 ) is parenthesis crossing is crossing corresponding \noindente lcorresponding lip ticTo comma this parabolic end we and define hyperbolic the K family hyphen orbits of double at points− withsided the r same i g h t value− angle of parameter cones phi b e open parametrised parenthesis 2 periodby $ 5 t >closing0e parenthesis l lip : $ tic ,.... parabolic of and hyperbolic K− orbits at points with the same value of parameter theφ subgroup(2.5) K period of \ beginFromthe{ thea subgroupl i gabove n ∗} algebraicK. and geometric descriptions of the orbits we can make several ob hyphen xservations ˆ{ 2 From} period+ the ( above y algebraic− \ f r a c and{ 1 geometric}{ 2 } descriptions( t + of t the ˆ{ orbits − 1 we} can) make ) ˆ{ several2 } − ob ( z − \ f r a c { 1 }{ 2 } (Remark t- − 2 periodt ˆ{ 7 period− 1 } ) ) ˆ{ 2 } = 0 . \ tag ∗{$ ( 2 . 7 ) $} \end1 period{servationsa l i g n ..∗} The . values of all three vector fields dK sub e comma dK sub p and dK sub h coincide on the quotedblleft real quotedblright U hyphenRemark axis v = 0 2 comma . 7 . \noindent The vertices of cones b elong t o the hyperbola $ \{ x = 0 , y ˆ{ 2 } − z ˆ{ 2 } i period e1 period . The they values are three of different all three extensions vector fields into thedK domaine, dKp ofand the samedKh coincide boundary on condition the “ realperiod ” U− =Another 1axis\} sourcev = 0,, of $ this see : .. Fig the axis . \ Uquad is the3 intersection for illustration of planes EE . to the power of prime comma PP to the power of prime and HH to the power ofi . prime e . they on are three different extensions into the domain of the same boundary condition \noindentFig. periodLemma 3 period $ 2 . 6 . K − $ orbits may be obtained cases as fo l lows : Another source of this : the axis U is the intersection of planes EE0,PP 0 and HH0 on \ centerline {( e ) . \quad e l l i p t i c $ KFig− . 3$ . orbits are s ections of cones ( 2 . 7 ) \quad by the plane $ z = 0 ( EE ˆ{\prime }$ on Fig . \quad 3 ) ; }

\ centerline {( p ) . \quad parabolic $ K − $ orbits are s ections of ( 2 . 7 ) \quad by the p lane $ y = \pm z ( PP ˆ{\prime }$ on Fig . 3 ) ; }

\ centerline {( h ) . \quad hyperbolic $ K − $ orbits are s ections of ( 2 . 7 ) \quad by the p lane $ y = 0 ( HH ˆ{\prime }$ on Fig . \quad 3 ) . }

\ hspace ∗{\ f i l l }Moreover , each straight line generating a cone from the family ( 2 . 7 ) is crossing corresponding

\noindent e l lip tic , parabolic and hyperbolic $ K − $ orbits at points with the same value of parameter $ \phi ( 2 . 5 ) $ \ h f i l l o f

\noindent the subgroup $K . $

\ hspace ∗{\ f i l l }From the above algebraic and geometric descriptions of the orbits we can make several ob −

\noindent servations .

\noindent Remark 2 . 7 .

\ hspace ∗{\ f i l l }1 . \quad The values of all three vector fields $ dK { e } , dK { p }$ and $ dK { h }$ coincide on the ‘‘ real ’’ $U − $ axis $v = 0 ,$

\ hspace ∗{\ f i l l } i . e . they are three different extensions into the domain of the same boundary condition .

\ hspace ∗{\ f i l l }Another source of this : \quad the axis $ U $ is the intersection of planes $ EE ˆ{\prime } , PP ˆ{\prime }$ and $ HH ˆ{\prime }$ on

\ centerline { Fig . 3 . } Erlangen Program at Large hyphen 1 : .... Geometry of Invariants .... 9 \noindenthlineErlangenErlangen Program Program at Large at - Large 1 :− 1 : \ h f i l l GeometryGeometry of of Invariants Invariants \ h f i l l 9 9 2 period .. The hyperbola passing through the point .. open parenthesis 0 comma 1 closing parenthesis has the shortest focal length square root\ [ \ ofr u 2 l among e {3em}{0.4 pt }\ ] all other hyperbolic orbits since it i s the section of the cone x to the power of 2 plus open parenthesis y minus 1 closing parenthesis to the power of√ 2 plus2 .z to the The power hyperbola of 2 = 0 passing closest through the point ( 0 , 1 ) has the shortest focal length \ hspacefrom2 the∗{\among familyf i l l } t2 o the . \ planequad HHThe to hyperbola the power of prime passing period through the point \quad ( 0 , 1 ) has the shortest focal length 2 2 2 $\ sqrt3 period{ 2 ..all} Two$ other among hyperbolas hyperbolic passing orbits through since open it iparenthesis s the section 0 comma of the v closing cone parenthesisx + (y − 1) and+ z open= 0 parenthesisclosest 0 comma v to the power 0 of minus 1 closing parenthesis have the samefrom focal the length family since t o they the plane HH . −1 \ hspaceare sections∗{\3 .f iof l l two Two} all cones hyperbolas other with the hyperbolic same passing distance through orbits from HH(0 since, v to) and the it power(0 i, v s of the) primehave section period the same Moreover of focal the comma length cone two since$ xsuch ˆ{ hyperbolas2 } + ( y − in1 thethey lower ) ˆ{ hyphen2 } and+ upper z ˆ{ half2 hyphen} = planes 0 $ passing c l o s e the s t points open parenthesis 0 comma v closing parenthesis and open parenthesis 0 0 comma minus v to theare power sections of minus of two 1 closing cones parenthesis with the aresame sections distance of from HH . Moreover , two such \ centerlinethehyperbolas same double{from hyphen the sided family cone t period o the .. They plane are related $ HH ˆ to{\ eachprime other} as explained. $ } in Remark 7 period 4 period 1 period −1 One canin see the from lower the - first and picture upper in half Fig period - planes 2 that passing the elliptic the points action of(0 subgroup, v) and K(0, fixes−v the) are sections of \ hspacepointthe e∗{\ sub samef1 i l period l double}3 . ..\ More -quad sided generallyTwohyperbolas cone . we have They : are passingthrough related to each other $( as 0 explained , v in )$ Remark and 7 . $( 0 , vˆ{ − 1 }Figure4) . $ 4 1 period . have .. the Actions same of the focal subgroups length which since fix point they e sub 1 in three cases period LemmaOne 2 period can 8see period from .. the The first fix group picture of the in point Fig . e 2 sub that 1 .. the is elliptic action of subgroup K fixes the \ hspaceopenpoint parenthesis∗{\ fe i1. l l }More eare closing sections generally parenthesis ofwe period twohave cones :.. the subgroup with the K sub same e to distance the power of from prime =$ KHH in ˆ the{\ eprime l lip tic} case. period $ Moreover .. Thus the , el two such hyperbolas liptic upper half hyphenFigure p lane is 4 a . modelActions for of the subgroups which fix point e1 in three cases . inthe theLemma homogeneous lower − 2 .and space 8 . upper SL subThe half 2 open fix− group parenthesisplanespassingthepoints of the R point closing parenthesise1 is slash $( K semicolon 0 , v )$ and $( 0 , − v ˆ{ − 1 } ) $ are sections0 of open parenthesis( e ) . p closingthe subgroup parenthesisKe period= K in .. the the subgroup e l lip tic N casesub p . to the Thus power the of prime el liptic of matrices upper half - p theEquation:lane same is double open a model parenthesis− forsided 2 period cone 8 . closing\quad parenthesisThey are ..Row related 1 1 0 Row to 2each nu e subother 0 1 . as = Rowexplained 1 0 e sub in 0 Row Remark 2 e sub 7 0 . 0 4 . Row. 1 1 . 1 nu e sub 0 Row 2 0 1 . Row 1 0 minus ethe sub 0homogeneous Row 2 minus e space sub 0 0SL . 2(R)/K; One can see from the first picture in Fig . 2 that0 the elliptic action of subgroup $ K $ fixes the in the parabolic case period .. It also( p )fixes . any the point subgroup ve 1 periodN ..p of It is matrices conjugate to subgroup N comma .. thus the pointparabolic $ upper e { half1 } hyphen. $ plane\quad is a modelMore for generally the homogeneous we have space : SL sub 2 open parenthesis R closing parenthesis slash N semicolon open parenthesis h closing parenthesis period .. the subgroup  A sub h to the  power of prime of matrices 1 0 0 e0 1 νe0 0 −e0 \ centerlineEquation: open{ Figure parenthesis 4 . 2\quad periodActions 9 closing= parenthesis of the subgroups ..Row 1 hyperpolic which cosine fix open point parenthesis $ e { tau1 closing}$(2.8) in parenthesis three caseshyperbolic . } sine νe0 1 e0 0 0 1 −e0 0 open parenthesis tau closing parenthesis e sub 0 Row 2 minus hyperbolic sine open parenthesis tau closing parenthesis e sub 0 hyperpolic cosine \noindent Lemma 2 . 8 . \quad The fix group of the point $ e { 1 }$ \quad i s open parenthesisin the tauclosing parabolic parenthesis case . . =1 It divided also fixes by 2 any Row point 1 1 minusve e1. subIt 0 Row is conjugate 2 minus e to sub subgroup 0 1 . Row 1N, e to the power of tau 0 Row 2 0 e tothus the power the of minus tau . Row 1 1 e sub 0 Row 2 e sub 0 1 . comma \ hspacein the∗{\ hyperbolicf i l l }( case e period ) . \quad .. It isthe conjugate subgroup to subgroup $ K A ˆ comma{\prime thus} two{ copiese } of= the K$ upper halfplane in the e l lip tic case . \quad Thus the el liptic upper half − p lane is a model for parabolic upper half - plane is a model for the homogeneous space SL2(R)/N; ( h ) . the open parenthesis s0 ee Section 7 period 2 closing parenthesis .. is a model for SL sub 2 open parenthesis R closing parenthesis slash A period subgroup Ah of matrices \ centerlineMoreover comma{ the vectors homogeneous fields of these space actions $ SL are open{ 2 parenthesis} ( R u to ) the /power K of 2 ; plus $ sigma} open parenthesis v to the power of 2 minus 1 closing parenthesis comma 2 uv closing parenthesis for the corresponding values \ centerline {( p ) . \quad the subgroup $ N ˆ{\prime } {τ p }$ of  matrices } of sigma period .. Orbitscosh( ofτ) the fix sinh( groupsτ)e0 satisfy to1 the1 equation−e0 : e 0 1 e0 = −τ , (2.9) parenleftbig u to the− powersinh(τ) ofe0 2 minuscosh(τ sigma) v to the2 power−e0 of1 2 parenrightbig0 e minus 2 lve minus0 1 sigma = 0 comma where l in R period \ beginRemark{ a l 2 i g period n ∗} 9 period \ l e1 fperiod t (\ beginin .. the Note{ array hyperbolic that}{ wecc can} case uniformly1 &. 0 express\\\ It isnu conjugatethe fixe hyphen{ to0 subgroups subgroup} & 1 \ ofend eA, sub{thusarray 1 in two all}\ right EPH copies cases) of = by the matrices\ l upper e f t (\ begin { array }{ cc } 0 & e { 0 }\\ e { 0 } & 0 \end{ array }\ right ) \ l e f t (\ begin { array }{ cc } 1 & \nu e { 0 }\\ 0 & 1 \end{ array }\ right ) \ l e f t (\ begin { array }{ cc } 0 of thehalfplane form : ( s ee Section 7 . 2 ) is a model for SL2(R)/A. 2 2 & Row− 1eMoreover a minus{ 0 sigma}\\ , vectors e− sub 0e fields b Row{ 0 of 2} these minus& 0 actions e\ subend 0{ barray are a . comma(}\u right+σ where(v −)\1)tag a, to2uv∗{ the)$for power ( the 2 of corresponding 2 minus . 8 sigma ) values b $ to} the power of 2 = 1 period \end{ofa l i g nσ.∗} Orbits of the fix groups satisfy to the equation :

2 2 \ hspace ∗{\ f i l l } in the parabolic(u case− σv ) .−\2quadlv − σ =It 0, also where fixes l ∈ R any. point $ ve 1 . $ \quad It is conjugate to subgroup $ N , $ \quad thus the Remark 2 . 9 . \noindent 1parabolic . Note upperthat we half can− uniformlyplane is express a model the fixfor - thesubgroups homogeneous of e1 in space all EPH $ cases SL { by2 } (R)/ N ;matrices $ ( h ) . \quad the subgroup $ A ˆ{\primeof the} form{ h :}$ of matrices  a −σe b  \ begin { a l i g n ∗} 0 , where a2 − σb2 = 1. −e b a \ l e f t (\ begin { array }{ cc }\cosh ( 0 \tau )& \sinh ( \tau ) e { 0 }\\ − \sinh ( \tau ) e { 0 } & \cosh ( \tau ) \end{ array }\ right ) = \ f r a c { 1 }{ 2 }\ l e f t (\ begin { array }{ cc } 1 & − e { 0 }\\ − e { 0 } & 1 \end{ array }\ right ) \ l e f t (\ begin { array }{ cc } e ˆ{\tau } & 0 \\ 0 & e ˆ{ − \tau }\end{ array }\ right ) \ l e f t (\ begin { array }{ cc } 1 & e { 0 }\\ e { 0 } & 1 \end{ array }\ right ), \ tag ∗{$ ( 2 . 9 ) $} \end{ a l i g n ∗}

in the hyperbolic case . \quad It is conjugate to subgroup $ A , $ thus two copies of the upper halfplane ( s ee Section 7 . 2 ) \quad is a model for $ SL { 2 } ( R ) / A . $

Moreover , vectors fields of these actions are $ ( u ˆ{ 2 } + \sigma ( v ˆ{ 2 } − 1 ) , 2 uv ) $ for the corresponding values o f $ \sigma . $ \quad Orbits of the fix groups satisfy to the equation :

\ [ ( u ˆ{ 2 } − \sigma v ˆ{ 2 } ) − 2 l v − \sigma = 0 , where l \ in R. \ ]

\noindent Remark 2 . 9 .

\ hspace ∗{\ f i l l }1 . \quad Note that we can uniformly express the fix − subgroups of $ e { 1 }$ in all EPH cases by matrices

\ centerline { o f the form : }

\ [ \ l e f t (\ begin { array }{ cc } a & − \sigma e { 0 } b \\ − e { 0 } b & a \end{ array }\ right ), where a ˆ{ 2 } − \sigma b ˆ{ 2 } = 1 . \ ] 1 0 .... V period V period Kisil \noindenthline1 0 1 0 \ h f i l l V . V . K i s i l V . V . Kisil 2 period .. In the hyperbolic case the subgroup A sub h to the power of prime may be extended t o a subgroup A sub h to the power of prime\ [ \ r prime u l e {3em by the}{0.4 element pt }\ ]

Row 1 0 e sub 0 Row 2 e sub 0 0 . comma which flips upper0 and lower half hyphen planes .. open parenthesis00 see Section .. 7 period 2 closing parenthesis2 . In period the hyperbolic.. The subgroup case A the sub subgroup h to the powerAh ofmay prime be prime extended t o a subgroup Ah by the \ hspaceelement∗{\ f i l l }2 . \quad In the hyperbolic case the subgroup $ A ˆ{\prime } { h }$ may be extended t o a subgroup fixes the set open0 e brace e sub 1 comma minus e sub 1 closing brace period $ A ˆ{\prime \0prime, which} { flipsh }$ upper by and the lower element half - planes ( see Section 7 . 2 ) . The Lemma 2 periode 100 period .... M dieresis-o bius action of SL sub 2 open parenthesis R closing parenthesis in each EPH case is generated by action the correspon0 hyphen subgroup A00 \ hspaceding fix∗{\ hyphenf i l l } subgrouph$\ l e f t open(\ begin parenthesis{ array A}{ subcc } h to0 the & power e { of0 prime}\\ primee { in0 the} hyperbolic& 0 \end case{ array closing}\ parenthesisright ) .... , $ and which actions flips upper and lower half − planes \quad ( see Section \quad 7 . 2 ) . \quad The subgroup fixes the set {e , −e }. of$ Athe ˆ ax{\ plusprime b group\ commaprime ....} { e periodh }$ g period subgroups A 1 1 Lemma 2 . 10 . M o¨ bius action of SL2(R) in each EPH case is generated by action the andcorrespon N period - \ centerlineProof period{ ....fixes The theax plus set b group $ \{ transitivelye { acts1 } ...., on the− uppere { or lower1 }\} half hyphen. $ plane} period .... Thus for any ding fix - subgroup (A00 in the hyperbolic case ) and actions of the ax + b group , e . g . g in SL sub 2 open parenthesish R closing parenthesis there i s h in ax plus b group such that h to the power of minus 1 g either fixes e sub subgroups A 1\noindent or sends it tLemma o minus 2 e .sub 10 1 period . \ h f .... i l l ThusM $ \ddot{o} $ bius action of $ SL { 2 } ( R ) $ in each EPH case is generated by action the correspon − and N. h to the power of minus 1 g i s in the corresponding fix hyphen group period blacksquare Proof . The ax + b group transitively acts on the upper or lower half - plane . Thus for \noindent2 period 3ding .. Invariance f i x − ofsubgroup cycles $ ( Aˆ{\prime \prime } { h }$ in the hyperbolic case ) \ h f i l l and actions of the $axany + b$ group, \ h f i l l e . g . subgroups $A$ As .. we will .. see .. soon the three .. typ es .. of K hyphen orbits−1 .. are .. principal invariants .. of the .. constructed g ∈ SL2(R) there i s h in ax + b group such that h g either fixes e1 or sends it t o −e1. Thus geometries−1 comma thus we will unify them in the following definition period \noindenth g iand s in the $ N corresponding . $ fix - group .  Definition2 . 3 2 period Invariance 1 1 period of .. We cycles use the word cycle t o denote loci in R to the power of sigma defined by the equation : Equation: open parenthesis 2 period 1 0 a closing parenthesis .. minus k open parenthesis ue sub 0 plus ve sub 1 closing parenthesis to the \noindentAsProof we will . \ h see f i l l soonThe the $ ax three + typ b $ es group of K transitively− orbits are acts principal\ h f i l l invariantson the upper or lower half − plane . \ h f i l l Thus f o r any power ofof 2 the minus constructed2 angbracketleft geometries open parenthesis , thus l comma we will n closingunify them parenthesis in the comma following open parenthesis definition u . comma v closing parenthesis right angbracket plus m = 0 \noindentDefinition$ g 2\ .in 1 1 .SL {We2 use} the( word Rcycle )$ thereist o denote loci $h$ in Rσ defined in $ax by the + equation b$ groupsuchthat $hˆ{ − or equivalently: open parenthesis avoiding any reference t o Clifford algebra generators closing parenthesis : 1 }Equation:g $ either open parenthesis fixes 2 $ period e { 11 0} b$ closing or sends parenthesis it to .. k $parenleftbig− e { u1 to} the. power $ \ ofh 2f i minusl l Thus sigma v to the power of 2 parenrightbig minus 2 lu minus 2 nv plus m = 0 comma where sigma = 2 e sub 1 comma \noindent $ h ˆ{ − 1 } g $ i s in the corresponding fix − group $ . \ blacksquare $ or equivalently open parenthesis using only Clifford algebra2 operations comma cf period open square bracket 7 2 comma Supplement C open −k(ue0 + ve1) − 2h(l, n), (u, v)i + m = 0 (2.10a) parenthesis 42 a closing parenthesis closing square bracket closing parenthesis : \noindent 2 . 3 \quad Invariance of cycles Lineor 1 equivalentlyKw to the power ( avoiding of 2 plus Lw any minus reference wL plus t oM Clifford = 0 comma algebra open parenthesis generators 2 period) : 1 0 c closing parenthesis Line 2 where w = ue sub 0 plus ve sub 1 comma K = minus ke sub 1 comma L = minus ne sub 0 plus l e sub 1 comma M = me sub 0 1 period \noindentSuch cyclesAs obviously\quad meanwe w for i l l certain\quad k commasee \quad l commasoon n comma the three m straight\quad linestyp and one es of\quad the followingo f $ : K − $ o r b i t s \quad are \quad principal invariants \quad o f the \quad constructed geometriesopen parenthesis , thus e closing we parenthesis will unify2 period2 them .. in in the the elliptic following case : .. circles definition with centre . parenleftbig l divided by k sub comma n divided k(u − σv ) − 2lu − 2nv + m = 0, where σ = 2e1 , (2.10b) by k to the power of parenrightbig and squared radius m minus l to the power of 2 plus n to the power of 2 divided by k semicolon \noindentopenor parenthesis equivalentlyDefinition p closing ( using 2 parenthesis .only 1 1 .Clifford period\quad ..We algebra in theuse parabolic operations the word case ,cycle : cf parabolas . [ 7t 2 o , with denote Supplement horizontal loci C directrix in ( 42 $ a R and ) ˆ]{\ )focussigma at open}$ parenthesis defined l by the equation : divided: by k sub comma m divided by 2 n minus l to the power of 2 divided by 2 nk plus n divided by 2 k closing parenthesis semicolon \ beginopen{ parenthesisa l i g n ∗} h closing parenthesis period .. in the hyperbolic case : .. rectangular hyperbolas with centre parenleftbig l divided by k sub− commak minus ( ue n divided{ 0 by} k+ to the ve power{ 1 of} parenrightbig)Kw ˆ{ 2 2+ Lw} .. and− wL a2 vertical+ M =\ langle 0 axis, (2 of.10c)( l , n ) , ( u , v ) \rangle + m = 0 \ tag ∗{$ ( 2 . 1 0 a ) $} symmetry period wherew = ue0 + ve1,K = −ke1,L = −ne0 + le1,M = me01. \endMoreover{ a l i g n words∗} parabola and hyperbola in this paper always assume only the above described typ esSuch period cycles .. Straight obviously lines are mean also calledfor certain flat cyclesk, l, period n, m straight lines and one of the following : \noindent or equivalently ( avoiding any reference t ol n Clifford) algebra generatorsl2+n2 ) : All three( e EPH ) . types in the of cycles elliptic are enjcase oying : many circles common with propertiescentre ( k comma, k and sometimes squared even radius beyondm − k ; that we normally expect( p ) . period in .. the For parabolic example comma case : the parabolas following definitionwith horizontal i s quite directrixintelligible and even focus when at \ begin { a l i g n ∗}2 extended( l m − froml the+ n above); elliptic and hyperbolic cases to the parabolic one period kDefinition (k , 2n u 2 ˆ2 periodnk{ 2 2} 1k 2 − period \sigma ring-sigmav hyphen ˆ{ 2 Centre} ) of the− sigma2 hyphen lu − cycle2 open nvparenthesis + 2m period = 1 0 0 closing , parenthesis where for\sigma any = 2 { e( h{ ) .1 }} in the, \ tag hyperbolic∗{$ ( case 2 : . rectangular 1 0 b hyperbolas ) $} with centre ( l − n ) and a EPH case i s the point parenleftbig l divided by k sub comma minus sigma-ring n divided by k parenrightbigk , k in Case 1 sigma Case 2 period \endNotions{verticala l i g ofn ∗} e hyphenaxis of centre comma p hyphen centre comma h hyphen centre are used along the adopted EPH notations period Centres of straight lines are at infinity comma seesymmetry Subsection 7 . period 1 period \noindentRemarkMoreover ..or 2 period equivalently words 1 3parabola period .. ( Hereand using wehyperbola use only a signature Cliffordin this ring-sigma paper algebra always = minus operations assume 1 comma only 0 , or the cf .. above1 . .. [ of 7 a described Clifford2 , Supplement algebra which C i ( s not 42 a ) ] ) : relatedtyp t es o the . signature Straight sigma lines of are the alsospace called R to theflat power cycles of sigma. period We will need also a third signature sigma-breve t o describ e the \ [ \geometrybeginAll{ a of l threei gcycles n e d } EPH inKw Definition ˆtypes{ 2 } 3 of period+ cycles 1 Lw period are− enj oyingwL+M=0,(2.10c) many common properties , sometimes even \\ whereThebeyond meaningfulness w that = we of ue normally this{ definition0 } expect+ even ve. in the For{ parabolic1 example} , case K, the is justified = following− commake definition for{ example1 } i s, quite comma L intelligible by = : − ne { 0 } + l e { even1 } when, M extended = me from{ the0 above 1 } elliptic. \end{ anda l i g hyperbolic n e d }\ ] cases to the parabolic one . Definition 2.12. ˚σ− Centre of the σ− cycle ( 2 . 1 0 ) for any EPH case i s the point σ \ centerline( l − ˚σ n{)Such∈ R cycles obviously mean for certain $ k , l , n , m$ straight lines and one of the following : } k , k . \ centerlineNotions{( of e e - ) centre . \quad , p -in centre the, elliptic h - centre case are used : \quad alongcircles the adopted with EPH centre notations $ ( .\ f r a c { l }{ k } { , }\ f r a c { n }{ k }ˆ{ ) }$ and squared radiusCentres $ m of− straight \ f r a c { linesl ˆ are{ 2 at} infinity+ n , ˆ see{ 2 Subsection}}{ k } 7; . $ 1 . } Remark 2 . 1 3 . Here we use a signature ˚σ = −1, 0 or 1 of a Clifford σ \ hspacealgebra∗{\ f i which l l }( p i s ) not . \quad relatedin t the o the parabolic signature caseσ of the : parabolas space R . withWe will horizontal need alsoa directrix third and focus at $ ( \ f r a c signature{ l }{ k σ˘} t{ o, describ}\ f r a e c the{ m geometry}{ 2 n of} cycles − \ inf r a Definition c { l ˆ{ 2 3}}{ . 1 . 2 nk } + \ f r a c { n }{ 2 k } ) ; $ The meaningfulness of this definition even in the parabolic case is justified , for example , \ hspaceby∗{\ : f i l l }( h ) . \quad in the hyperbolic case : \quad rectangular hyperbolas with centre $ ( \ f r a c { l }{ k } { , } − \ f r a c { n }{ k }ˆ{ ) }$ \quad and a vertical axis of

\ centerline {symmetry . }

\noindent Moreover words parabola and hyperbola in this paper always assume only the above described typ es . \quad Straight lines are also called flat cycles .

All three EPH types of cycles are enj oying many common properties , sometimes even beyond that we normally expect . \quad For example , the following definition i s quite intelligible even when extended from the above elliptic and hyperbolic cases to the parabolic one .

\noindent Definition $2 . 1 2 . \ mathring {\sigma} − $ Centre of the $ \sigma − $ cycle ( 2 . 1 0 ) for anyEPH case i s the point $\ l e f t .( \ f r a c { l }{ k } { , } − \ mathring {\sigma}\ f r a c { n }{ k } ) \ in R\ begin { a l i g n e d } & \sigma \\ &. \end{ a l i g n e d }\ right . $

\noindent Notions o f e − c e n t r e , p − c e n t r e , h − centre are used along the adopted EPH notations .

\ centerline { Centres of straight lines are at infinity , see Subsection 7 . 1 . }

\noindent Remark \quad 2 . 1 3 . \quad Here we use a signature $ \ mathring {\sigma} = − 1 , 0 $ or \quad 1 \quad of a Clifford algebra which i s not related t o the signature $ \sigma $ of the space $Rˆ{\sigma } . $ We will need also a third signature $ \breve{\sigma} $ t o describ e the geometry of cycles in Definition 3 . 1 .

\ hspace ∗{\ f i l l }The meaningfulness of this definition even in the parabolic case is justified , for example , by : Erlangen Program at Large hyphen 1 : .... Geometry of Invariants .... 1 1 \noindenthlineErlangenErlangen Program Program at Large at - Large 1 :− 1 : \ h f i l l GeometryGeometry of of Invariants Invariants \ h f i l l 1 1 1 1 bullet .. the uniformity of description of relations b etween centres of orthogonal cycles comma see the next \ [ subsection\ r u l e {3em and}{ Fig0.4 period pt }\ 8] period bullet .. the appearance of concentric parabolas in Fig period 1 7 open parenthesis N sub P sub e closing parenthesis and open parenthesis N sub P sub h• closingthe parenthesis uniformity period of description of relations b etween centres of orthogonal cycles , see \ hspaceUsingthe the∗{\ next Lemmasf i l l } $ 2 period\ bullet 3 and$ 2 period\quad 5 wethe can uniformity give an easy open of description parenthesis and of virtually relations calculation b etween hyphen free centres ! closing of parenthesis orthogonal cycles , see the next .. proof of subsection and Fig . 8 . \ centerline { subsection• the appearance and Fig of . 8concentric . } parabolas in Fig . 17(N ) and (N ). invariance for corresp onding cycles period Pe Ph LemmaUsing 2 period the 14 Lemmas period .. M 2 .dieresis-o 3 and 2 bius . 5 transformations we can give an preserve easy ( the and cycles virtually in the upper calculation half hyphen - free plane ! ) comma i period e period : \ centerlineopenproof parenthesis of{ $ \ ebullet closing parenthesis$ \quad periodthe.. appearance For C-lscript of open concentric parenthesis e parabolas closing parenthesis in Fig M o-dieresis $ . 1 biustransformations 7 ( N { mapP { e }} circles) $invariance toand circles $ period( for N corresp{ P { ondingh }} cycles) . . $ } openLemma parenthesis 2 . p 14 closing . parenthesisM o¨ bius period transformations .. For C-lscript open preserve parenthesis the cycles p closing in the parenthesis upper half M dieresis-o - plane bius transformations map parabolas\ hspace, i∗{\ to. e parabolasf . i :l l } Using period the Lemmas 2 . 3 and 2 . 5 we can give an easy ( and virtually calculation − f r e e ! ) \quad proof o f open parenthesis( e ) . h closing For parenthesisC − lscript period(e) M .. Foro¨ bius C-lscript transformations open parenthesis map h closing circles parenthesis to circles M . o-dieresis bius transformations map hyperbolas\noindent( to pinvariance )hyperbolas . For period forC − lscript corresp(p) M ondingo¨ bius cycles transformations . map parabolas to parabolas . ( h ) . For C − lscript(h) M o¨ bius transformations map hyperbolas to hyperbolas . Figure 5 period .. Decomposition of an arbitrary M dieresis-o bius transformation g into a product g = g a0 g0 n g k g sub a to the power of prime\noindent g subFigure nLemma to 5 the . power 2Decomposition . 14 of period . \quad to of theM an power arbitrary $ \ddot of prime M{oo ¨}bius$ transformation bius transformationsg into a product preserveg = gagngkg theagn cycles. in the upper half − plane , i . e . : ProofProof period . .... Our first first .... observation i s i that s that .... the the .... subgroups ....A ....and andN Nobviously .... obviously preserve preserve all all circles comma \ centerlineparabolascircles comma ,{( e hyperbolas ) . \quad andFor straight $ C− lineslscript in all C l open( eparenthesis )$M$ sigma closing\ddot{ parenthesiso} $ bius period transformations .... Thus we use subgroups map circles A to circles . } and Nparabolas to fit , hyperbolas and straight lines in all C`(σ). Thus we use subgroups A and N to fit \ centerlinea givena given cycle{ cycle( exactly p ) exactlyon . \ aquad particular onFor a particular orbit $ C of−lscript subgroup orbit Kof shown (subgroup p on Fig )$K periodshown M 2 $ of on\ theddot Fig corresp{o .} 2 onding$ of bius the corresp transformations map parabolas to parabolas . } onding typ e . typ e period 0 \ centerlineTo this{( hend ) for . \ anquad arbitraryFor $ cycle C−lscriptS we can (find hgn )$∈ N Mwhich $ \ddot puts{ centreo} $ of biusS on transformations the map hyperbolas to hyperbolas . } To this end for an arbitrary cycle S we can find g sub n to the0 power of prime in N which puts centre of S on the V hyphenV − axis axis , comma see Fig .... . see 5 . Fig Then period there .... 5 isperiod a unique .... Thenga there∈ isA awhich unique gscales sub a it to exactly the power t ofo anprime orbit in A which scales it exactly t \ centerlineof K, { Figure 5 . \quad Decomposition of an arbitrary M $ \ddot{o} $ bius transformation $ g $ into a product o an orbit of K comma 1 $g=gagngkgˆe . g . for a circle passing through points{\prime(0, v1) and} { (0a, v}2) theg ˆ scaling{\prime factor} { i sn√ ˆ{ .according}}$ } e period g period for a circle passing through points open parenthesis 0 comma v sub 1 closing parenthesisv1v2 and open parenthesis 0 comma v sub 2 closingt o parenthesis the scaling factor i s 1 divided by square root of v sub 1 v sub 2 according t o \noindent Proof . \ h f i l l Our f i0 r s t \0 h0 f i l l observation i s that \ h f i l l the \ h f i l l subgroups \ h f i l l $ A $ LemmaLemma 2 period 2 . 5 5 open ( e parenthesis ) . Let eg closing= gagn parenthesis, then for period any element Let g to theg ∈ powerSL2( ofR) primeusing = the g sub Iwasawa a to the power of prime g sub n to\ h the f i ldecomposition powerl and of comma$ N $ to\ theh f i power l l obviously of prime then preserve for any element all circles g in SL sub , 2 open parenthesis R closing parenthesis using the Iwasawa 0−1 0 0 0 0 decompositionof gg = gagngk we get the presentation g = gagngkgagn with ga, ga ∈ A, gn, gn ∈ N \noindentof ggand to theparabolas power of prime , hyperbolas minus 1 = g and a g n straight g k we get the lines presentation in all g =$ Cg a g\ ne g l lk g sub( a to\sigma the power) of prime . $ g sub\ h f n i l to l theThus we use subgroups power$A$ of prime and with $N$ g a comma to fit g sub a to the power of prime in A comma g n comma g sub n to the power of prime in N and g k in K period \noindentThen the imagea given g to the cycle power exactly of prime S on of athe particular cycle Sgk under∈ K. gorbit to the ofpower subgroup of prime = $ g subK $ a to shown the power on of Fig prime . 2g sub of n the to the corresp power onding typ e . of prime .. i s .. a cycle it self in the .. obvious0 way comma 0 0 0 Then the image g S of the cycle S under g = gagn i s a cycle it self in the thenobvious g k open way parenthesis , g to the power of prime S closing parenthesis .... is again a cycle since g to the power of prime S was arranged t o coincide\ hspace with∗{\ af i K l l hyphen}To this orbit end comma for .... an and arbitrary finally cycle $ S $ we can find $ g ˆ{\prime } { n }\ in N $ whichthen putsgk centre(g0S) ofis again $ S a $ cycle on sincethe g0S was arranged t o coincide with a K− orbit , and gS =finally g a g n open parenthesis g k open parenthesis g to the power of prime S closing parenthesis closing parenthesis i s a cycle due t o the obvious action of g a g n comma see Fig period 5 .... for an illustration period blacksquare gS = gagn(gk(g0S)) i s a cycle due t o the obvious action of gagn, see Fig . 5 for an illustration \noindentOne can naturally$ V wish− $ that a x all i s other , \ h proofs f i l l insee this Fig paper . will\ h b f i e l ofl the5 . same\ h f sort i l l periodThen .. there This i sis a unique $ g ˆ{\prime } { a } \ in . A$ which scales it exactly t o an orbit of $K , $ likely tOne o be can possible naturally comma wish however that we all use other a lot of proofs computer in this algebra paper calculations will b e of asthe well same period sort . This 3 ..i Space s likely .. of t cycles o be possible , however we use a lot of computer algebra calculations as well . \noindentWe saw ine the . previous g . for sections a circle that cycles passing are M through o-dieresis bius points invariant $ (comma 0 thus , they v are{ natural1 } obj) $ ects and $ ( 0 , v { 2 } ) $of the3 the corresponding scaling Space factor geometries of cycles i sin the $\ sensef r a c { of1 F period}{\ sqrt Klein{ v period{ 1 ..} Anv efficient{ 2 tool}}} of$ their according study i s t t o o o¨ representWe saw all cyclesin the in previous R to the powersections of sigma that bycycles points are of M a newbius bigger invariant space period , thus they are natural obj \noindentects ofLemma2 the corresponding . 5 ( e ) geometries . Let $gˆ in{\ theprime sense of} F= . Klein g ˆ{\ .prime An efficient} { a } toolg of ˆ{\ theirprime } { n ˆ{ , }}$ 3 period 1 .. Fillmore endash Springer endash Cnopsσ construction .. open parenthesis FSCc closing parenthesis thenIt ....study for is well any i s knownt element o represent that linear $ g all hyphen cycles\ in fractional inSL R { transformationsby2 points} ( of R a .... new ) can $ bigger b e using linearised space the. by Iwasawa a transition decomposition into a suitable3 . 1 proj Fillmore ective space – .... Springer open square – bracket Cnops 6 0 construction comma .... Chapter (1 closing FSCc square ) bracket period .... The fundamental idea of the Fillmore\noindentIt endash iso well f Springer $ known gg endash ˆ{\ thatprime linear− - fractional1 } = transformations g a g n can g b e linearised k$ wegetthepresentation by a transition $g = g aCnops ginto construction n g k open g parenthesis ˆ{\prime FSCc} { closinga } parenthesisg ˆ{\prime open square} { n bracket}$ with 1 .... 6 comma $g .... a 6 3 , closing gˆ square{\prime bracket} i{ s thata }\ for in Alinearisation ,a suitable g of M n o-dieresis proj , ective g bius ˆ{\ space transformationprime [ 6} 0{ , inn RChapter}\ to thein power 1N ] . of$ sigma The and fundamental the idea of the Fillmore – Springer – \ beginCnops{ a l i g nconstruction∗} ( FSCc ) [ 1 6 , 6 3 ] i s that for linearisation of M o¨ bius gtransformation k \ in K. in Rσ the \end{ a l i g n ∗}

\ hspace ∗{\ f i l l }Then the image $ g ˆ{\prime } S$ of the cycle $S$ under $gˆ{\prime } = g ˆ{\prime } { a } g ˆ{\prime } { n }$ \quad i s \quad a cycle it self in the \quad obvious way ,

\noindent then $ g k ( g ˆ{\prime } S ) $ \ h f i l l is again a cycle since $ g ˆ{\prime } S $ was arranged t o coincide with a $ K − $ o r b i t , \ h f i l l and f i n a l l y

\noindent $gS=gagn ( gk ( gˆ{\prime } S ) ) $ i s a cycle due t o the obvious action of $g a g n ,$ seeFig.5 \ h f i l l for an illustration $ . \ blacksquare $

One can naturally wish that all other proofs in this paper will b e of the same sort . \quad This i s likely t o be possible , however we use a lot of computer algebra calculations as well .

\noindent 3 \quad Space \quad o f c y c l e s

\noindent We saw in the previous sections that cycles are M $ \ddot{o} $ bius invariant , thus they are natural obj ects of the corresponding geometries in the sense of F . Klein . \quad An efficient tool of their study i s t o represent all cycles in $ R ˆ{\sigma }$ by points of a new bigger space .

\noindent 3 . 1 \quad Fillmore −− Springer −− Cnops construction \quad ( FSCc )

\noindent I t \ h f i l l is well known that linear − fractional transformations \ h f i l l can b e linearised by a transition into

\noindent a suitable proj ective space \ h f i l l [ 6 0 , \ h f i l l Chapter 1 ] . \ h f i l l The fundamental idea of the Fillmore −− Springer −−

\noindent Cnops construction ( FSCc ) [ 1 \ h f i l l 6 , \ h f i l l 6 3 ] i s that for linearisation ofM $ \ddot{o} $ bius transformation in $ R ˆ{\sigma }$ the 1 2 .... V period V period Kisil \noindenthline1 2 1 2 \ h f i l l V . V . K i s i l V . V . Kisil required proj ective space can b e identified with the space of all cycles in R to the power of sigma period .. The latter can b e \ [ associated\ r u l e {3em with}{ certain0.4 pt }\ subset] of 2 times 2 matrices period FSCc can b e adopted from open square bracket 1 6 comma .. 6 3 closing square bracket t o serve all σ threerequired EPH cases proj with ective some space interesting can modifications b e identified period with the space of all cycles in R . The latter \noindentDefinitioncan b ....required e associated 3 period proj1 period with ective .... certain Let space.... subset P to can the of power b2 × e2 ofidentifiedmatrices 3 .... be .... . FSCc the with proj can the ective b space e .... adopted space of comma all from cycles .... [ 1 i 6 period , in e $ period R ˆ{\ ....sigma collection} . $ 6\quad 3 ] t oThe serve latter all three can EPH b e cases with some interesting modifications . .... of the rays .... passing through 3 associatedpointsDefinition in R to with the power certain3 . 1 of . 4 period subsetLet ....P of Webe .... $ the 2 define proj\times the ective following2 space $ two matrices identifications , i . e . . collection FSCc .... open can parenthesis ofb ethe adopted rays depending from from [ 1.... 6 some , \quad .... 6 3 ] t o serve all threepassing EPH cases through with some interesting modifications . additional 4 parameterspoints in sigmaR . commaWe define sigma-breve the following and s describ two ed identifications below closing parenthesis ( depending which from map some a point additional open parenthesis k comma l comma n \noindentparametersD e f i n iσ, t iσ˘ o nand\ hs f idescrib l l 3 . ed 1 below . \ h f )i l which l Let map\ h f i a l l point$ P(k, ˆ{ l, n,3 m})$∈ P\3hto f i :l l be \ h f i l l the proj ective \ h f i l l space , \ h f i l l i . e . \ h f i l l c o l l e c t i o n \ h f i l l o f the rays \ h f i l l passing through comma m closing parenthesis in P to the power of 3 to : σ Q : .. the cycleQ : openthe parenthesis cycle ( quadric quadric closing)C on parenthesisR defined C by on the R to equations the power ( of 2 sigma . 1 0 defined ) with by constant the equations open parenthesis 2 period\noindentparameters 1 0 closingpoints parenthesisk, in $Rˆ with constant{ 4 } parameters. $ \ h f k i l comma l We \ h f i l l define the following two identifications \ h f i l l ( depending from \ h f i l l some \ h f i l l a d d i t i o n a l l comma n comma m : Equation: open parenthesis 3 period 1 closing parenthesis .. minus k open parenthesis e sub 0 u plus e sub 1 v closing\noindent parenthesisparameters to the power $ of\sigma 2 minus 2 angbracketleft, \breve{\ opensigma parenthesis} $ and l comma $ s n $ closing describ parenthesis ed below comma ) open which parenthesis map a u point comma l, n, m : v$ closing ( k parenthesis , l right , angbracket n , plus m m = ) 0 comma\ in P ˆ{ 3 }$ to : 2 for some C-lscript open parenthesis sigma−k(e0 closingu + e1v parenthesis) − 2h(l, n), with(u, v) generatorsi + m = 0, 0 e to the power of 2 = minus(3. 11) comma 2 e sub 1 = sigma period\ hspace ∗{\ f i l l } $ Q : $ \quad the cycle ( quadric $ ) C$ on $Rˆ{\sigma }$ defined by the equations ( 2 . 1 0 ) with constant parameters $ kfor , some$ C − lscript(σ) with generators 02 = −1, 2 = σ. M : the ray of 2 × 2 matrices passing M : .. the ray of 2 times 2 matrices passing throughe e1 Equation:through open parenthesis 3 period 2 closing parenthesis .. C sub sigma-breve to the power of s = Row 1 l breve-e sub 0 plus sn e-breve sub 1\ begin m Row{ 2a l k i g minus n ∗} l breve-e sub 0 minus sn e-breve sub 1 . in M sub 2 open parenthesis C l open parenthesis breve-sigma closing parenthesis closingl ,parenthesis n , comma m with : \\ 0 e-breve − k to the ( power e  of{ 20 =} minusu 1 +comma e 2{ breve-e1 } subv 1 = ) sigma-breve ˆ{ 2 } − sub comma2 \ langle ( l , n ) , (s ule˘0 + ,sne˘ v1 ) m\rangle + m = 02 , \ tag ∗{$ ( 3 . 1 ) $} i period e period ..Cσ generators˘ = e-breve sub 0 .. and breve-e∈ M sub2(C 1` ..(˘σ of)), C-lscriptwith open0e˘ = − parenthesis1, 2e˘1 = breve-sigmaσ ˘, (3 closing.2) parenthesis .. can be .. k −le˘0 − sne˘1 of\end any{ ..a l type i g n ∗} : .. elliptic comma .. parabolic or hyperbolic

regardlessi . ofe . the C-lscript generators opene˘ parenthesis0 and e˘1 sigmaof closingC − lscript parenthesis(˘σ) can in open be parenthesisof any 3 period type 1: closing elliptic parenthesis period \noindentThe, meaningful parabolicf o r some values or hyperbolicof $ parameters C−l s c r i p sigmat ( comma\sigma sigma-breve) $ and with s are minus generators 1 comma 0 $ or 0 1 comma{ e }ˆ and{ 2 in} many= cases− s i1 s equal , 2 { e { 1 }} = \sigma . $ t oregardless sigma period of the C − lscript(σ) in ( 3 . 1 ) . The meaningful values of parameters σ, σ˘ and s $ M : $ \quad the ray o f $ 2 \times 2 $ matrices passing through Remarkare − 31 period, 0 or 1 2 , period and in A many hint for cases the compositions i s equal of the matrix open parenthesis 3 period 2 closing parenthesis is provided by the followingt o identityσ. : \ beginparenleftbigRemark{ a l i g n 1∗} w 3 parenrightbig . 2 . A Row hint 1 for L M the Row composition 2 K minus L . of Row the 1 w matrix Row 2 ( 1 .3 = . wKw 2 ) is plus provided Lw minus by wL the plus M comma Cwhich ˆ{followings realises} {\ thebreve identity equation{\sigma : open}} parenthesis= \ l e 2 f period t (\ begin 1 0 c{ closingarray }{ parenthesiscc } l of\breve a cycle{ periode} { 0 } + sn \breve{e} { 1 } & m \\The bothk & identifications− l \ Qbreve .. and{e} M ..{ are0 } straightforward − sn \ periodbreve ..{e Indeed} { comma1 }\end ..{ aarray point}\ ..right open parenthesis) \ in M k comma{ 2 } l comma(C n comma\ e l l m( closing\breve parenthesis{\sigma in P} to the) powerLM ) of , 3 with  w  0 {\breve{e}}ˆ{ 2 } = − 1 , 2 {\breve{e} { 1 }} (1 w) = wKw + Lw − wL + M, = equally\breve well{\ representssigma} open{ , parenthesis}\ tag ∗{K$ as (− soonL 3 as sigma .1 comma 2 ) sigma-breve $} and s are already fixed closing parenthesis both the equation open\end parenthesis{ a l i g n ∗} 3 period 1 closing parenthesis and the raywhich of matrix realises open parenthesis the equation 3 period ( 2 2. closing 1 0 c ) parenthesis of a cycle period . .. Thus for fixed sigma comma sigma-breve and s one can introduce the Q M correspondence\ hspace ∗{\ f i b l letween}Thei . both e . \ identificationsquad generatorsand $ \breveare{e} straightforward{ 0 }$ \quad .and Indeed $ \ ,breve a point{e} { 1 }$ \quad o f $ C−l s c r i p t (k, l, n, m) ∈ 3 ( quadrics\breve and{\sigma matrices}P shown) $ by\ thequad horizontalcan be arrow\quad on theo f following any \quad diagramtype : : \quad e l l i p t i c , \quad parabolic or hyperbolic σ, σ˘ s Quadricsequally d113-d113-d113-d113-d113-d113-d113-d113 well represents ( as soon as and onare R to already the power fixed of sigma ) both hline the sub equation Q circ M ( to3 . the 1 ) power and of Q P to the power of 3 σ, σ˘ s d73-d73-d73-d73-d73-d73-d73\noindentthe rayregardless of matrix ( of 3 to . the 2 ) power . $ C Thus− ofl M s c subr for i p tM fixed sub ( 2 open\andsigma parenthesisone) can $ to introduce the in power ( 3 .of the 1 C-lscript correspondence) . open parenthesis breve-sigma closing parenthesisTheb meaningful etween closing quadrics parenthesis values and open of matrices parameters parenthesis shown 3 period $ by\sigma the 3 closing horizontal parenthesis, \breve arrow{\ onsigma the following} $ and diagram $s$ : are $ − 1 , 0 $ or 1 , and in many cases $ s $ i s equalQuadrics d113 − d113 − d113 − d113 − d113 − d113 − d113 − which combines3 Q and M period .. On theM first glance the dotted arrow seems t o b e of a lit t le practical Q P d73−d73−d73−d73−d73−d73−d73 d113 σ Q◦M C−lscript (3.3) intereston sinceR it depends from t oo many differentM2( parameters(˘σ)) open parenthesis sigma comma sigma-breve and s closing parenthesis period However\noindentwhich the followingt combines o $ \sigmaQ and .M. $ On the first glance the dotted arrow seems t o b e of a lit t le resultpractical demonstrates interest that since it i s compatible it depends with from easy t calculationsoo many different of images parameters of cycles under(σ, theσ˘ and s). However \noindentM dieresis-othe followingRemark bius transformations 3 result . 2 demonstrates. A hintperiod for that the it composition i s compatible of with the matrix easy calculations ( 3 . 2 ) of is images provided of by the following identity : Propositioncycles under 3 period the 3 period M o¨ bius .... A transformations cycle minus k open . parenthesis e sub 0 u plus e sub 1 v closing parenthesis to the power of 2 minus 2 \ [ ( 1 w ) \ l e f t (\ begin { array }{ cc } L&M \\2 K& − L \end{ array }\ right ) \ l e f t (\ begin { array }{ c} w \\ angbracketleftProposition open parenthesis 3 . 3 . l comma nA closing cycle parenthesis−k(e0u + commae1v) − open2h(l, parenthesis n), (u, v)i + um comma= 0 is v transformed closing parenthesis by right angbracket plus m 1 \end{ array }\ right ) = wKw + Lw − wL + M , \ ] = 0 is transformedg ∈ SL2(R) by g in SL sub 2 open parenthesis R closing parenthesis ˜ 2 ˜l intointo the cycle the cycle minus k-tilde−k(e0u open+ e1 parenthesisv) − 2h( , n˜ e), sub(u, v 0)i u+ plusm ˜ = e0 subsuch 1 v that closing parenthesis to the power of 2 minus 2 angbracketleftBig open parenthesis to the power of l-tilde comma n-tilde closing parenthesis comma open parenthesis u comma v closing parenthesis angbracketrightBig plus\noindent tilde-m =which 0 such thatrealises the equation ( 2 . 1 0 c ) of a cycle . ˜s s −1 Equation: open parenthesis 3 period 4 closing parenthesisCσ˘ = gC ..σ˘ C-tildeg sub breve-sigma to the power of s = gC(3 sub.4) sigma-breve to the power of\ hspace s g to the∗{\ powerf i l l } ofThe minus both 1 identifications $ Q $ \quad and $ M $ \quad are straightforward . \quad Indeed , \quad a point \quad $ (for anyfor k Clifford any , Clifford algebras l , algebras C-lscript n , openC − m parenthesislscript ) (σ)\ in sigmaand closingPlscript ˆ{ 3 parenthesis−}$C(˘σ). ..Explicitly and lscript-C this open means parenthesis : sigma-breve closing parenthesis period .. Explicitly this means : \noindent equally well represents ( as soon as $ \sigma , \breve{\sigma} $ and $ s $ are already fixed ) both the equation ( 3 . 1 ) and the Equation: ˜l open+ sn parenthesis˜e˘ m˜ 3 period 5 closing parenthesisa be˘  ..Rowle˘ + 1 tilde-lsne˘ sub breve-em sub  0 plusd s− tilde-nbe˘  e-breve sub 1 tilde-m Row 2 k-tilde minusray l-tilde of matrix sube˘0 e-breve (1 3 sub . 2 0 minus ) . \ squad n-tilde= Thus breve-e for sub0 fixed 1 . = Row0 $ 1\ asigma b1 breve-e, sub 0\ Rowbreve 2 minus{\sigma c0 e-breve} .$ sub(3 and.5) 0 d $ . Row s $ 1 l one e-breve can sub introduce 0 the correspondence b etween k˜ −˜l − sn˜e˘ −ce˘0 d k −le˘0 − sne˘1 ce˘0 a plusquadrics sn breve-e and sub matrices1 m Rowe˘0 2 k shown minus1 l by e-breve the sub horizontal 0 minus sn breve-e arrow sub on 1 the . Row following 1 d minus b diagram breve-e sub : 0 Row 2 c e-breve sub 0 a . period \ hspace ∗{\ f i l l } Quadrics $ d113−d113−d113−d113−d113−d113−d113−d113 { on R ˆ{\sigma }\ r u l e {3em}{0.4 pt }}ˆ{ Q P ˆ{ 3 } d73−d73−d73−d73−d73−d73−d73 ˆ{ M }} { Q \ circ M } { M { 2 } ( ˆ{ C−l s c r i p t } ( \breve{\sigma} )) } ( 3 . 3 ) $

\noindent which combines $Q$ and $M . $ \quad On the first glance the dotted arrow seems t o b e of a lit t le practical interest since it depends from t oo many different parameters $ ( \sigma , \breve{\sigma} $ and $ s ) . $ However the following result demonstrates that it i s compatible with easy calculations of images of cycles under the M $ \ddot{o} $ bius transformations .

\noindent Proposition 3 . 3 . \ h f i l l A c y c l e $ − k ( e { 0 } u + e { 1 } v ) ˆ{ 2 } − 2 \ langle (l,n),(u,v) \rangle + m = 0$ is transformedby $g \ in SL { 2 } ( R ) $

\noindent into the cycle $ − \ tilde {k} ( e { 0 } u + e { 1 } v ) ˆ{ 2 } − 2 \ langle ( ˆ{\ tilde { l }} , \ tilde {n} ) , ( u , v ) \rangle + \ tilde {m} = 0$ such that

\ begin { a l i g n ∗} \ tilde {C} ˆ{ s } {\breve{\sigma}} = gC ˆ{ s } {\breve{\sigma}} g ˆ{ − 1 }\ tag ∗{$ ( 3 . 4 ) $} \end{ a l i g n ∗}

\noindent for any Clifford algebras $ C−l s c r i p t ( \sigma ) $ \quad and $ lscript −C( \breve{\sigma} ) . $ \quad Explicitly this means :

\ begin { a l i g n ∗} \ l e f t (\ begin { array }{ cc }\ tilde { l } {\breve{e} { 0 }} + s \ tilde {n}\breve{e} { 1 } & \ tilde {m}\\ \ tilde {k} & − \ tilde { l } {\breve{e} { 0 }} − s \ tilde {n}\breve{e} { 1 }\end{ array }\ right ) = \ l e f t (\ begin { array }{ cc } a & b \breve{e} { 0 }\\ − c \breve{e} { 0 } & d \end{ array }\ right ) \ l e f t (\ begin { array }{ cc } l \breve{e} { 0 } + sn \breve{e} { 1 } & m \\ k & − l \breve{e} { 0 } − sn \breve{e} { 1 }\end{ array }\ right ) \ l e f t (\ begin { array }{ cc } d & − b \breve{e} { 0 }\\ c \breve{e} { 0 } & a \end{ array }\ right ). \ tag ∗{$ ( 3 . 5 ) $} \end{ a l i g n ∗} Erlangen Program at Large hyphen 1 : .... Geometry of Invariants .... 1 3 \noindenthlineErlangenErlangen Program Program at Large at - Large 1 :− 1 : \ h f i l l GeometryGeometry of of Invariants Invariants \ h f i l l 1 3 1 3 Figure 6 period .. open parenthesis a closing parenthesis Different EPH implementations of the same cycles defined by quadruples of numbers period\ [ \ r u.. lopen e {3em parenthesis}{0.4 pt }\ b closing] parenthesis Centres and fo ci of two parabolas with the same fo cal length period ProofFigure period 6 .... . It( is a already ) Different established EPH implementations in the elliptic and of the hyperbolic same cycles cases defined for sigma by quadruples = sigma-breve of numbers sub comma . ( see open square bracket 1 6\noindent closingb ) square CentresFigure bracket and fo6 period ci . of\quad two .... parabolasFor( all a ) Differentwith the same EPH fo cal implementations length . of the same cycles defined by quadruples of numbers . \quad ( b ) CentresEPHProof cases and open . fo parenthesisIt ci is of already two including parabolas established parabolic with in closing the the elliptic parenthesis same and fo it hyperbolic cal can be length done cases by . the for directσ = calculationσ ˘, see [ 1 in 6 G ] . i NaC open square bracket 4 6 commaFor all.... S 2 period 7 closing square bracket period \noindentAnEPH alternative casesProof idea ( including. of\ anh f elegant i l l parabolicIt proof is alreadybased ) it on can the established be zero done hyphen by radiusthe indirect the cycles elliptic calculation and orthogonality and in hyperbolicG openi NaC parenthesis[ 4 cases 6 , see for $ \sigma = \breveb elow§2{\ . closingsigma 7 ] . } parenthesis{ , }$ may see be borrowed [ 1 6 ] from . \ openh f i l squarel For bracket a l l 1 6 closing square bracket period blacksquare ExampleAn alternative 3 period 4 period idea of an elegant proof based on the zero - radius cycles and orthogonality ( \noindent1 periodsee ..EPH The cases real axis ( vincluding = 0 i s represented parabolic by the ) it ray can coming be through done by .. openthe directparenthesis calculation 0 comma 0 comma in G 1i comma NaC [ 0 4 closing 6 , \ h f i l l \S 2 . 7 ] . parenthesisb elow .. and ) may a matrix be borrowed from [1 6].  \noindentRowExample 1 s e-breveAn alternative 3 sub . 41 0 . Row 2 idea 0 minus of s an breve-e elegant sub 1 . proof period based.. For any on .. the Row zero 1 a b− breve-eradius sub 0 cycles Row 2 minus and orthogonalityc e-breve sub 0 d . ( in see SL sub 2 open1 . parenthesis The real R closingaxis v parenthesis= 0 i s we represented have : by the ray coming through ( 0 , 0 , 1 , 0 ) \noindentand abelow matrix )maybeborrowedfrom $ [ 1 6 ] . \ blacksquare $ Row 1 a b breve-e sub 0s Rowe˘ 20 minus c e-breve sub 0 d . Rowa 1 b se˘ e-breve sub 1 0 Row 2 0 minus s breve-e sub 1 . Row 1 d minus b breve-e 1 . For any 0 ∈ SL ( ) we have : sub 0 Row 2 c e-breve sub 00 a . =−s Rowe˘ 1 s e-breve sub 1 0 Row−ce˘ 2 0d minus s breve-e2 R sub 1 . comma \noindenti period e periodExample .. the 3 real . 4 line . is1 SL sub 2 open parenthesis0 R closing parenthesis hyphen invariant period  a be˘   se˘ 0   d −be˘   se˘ 0  2 period .. A .. direct .. calculation0 .. in .. GiNaC1 .. open square bracket0 4 ..= 6 comma1 .. S .. 3, period 2 period 1 closing square bracket .. shows\ hspace .. that∗{\ ..f i matrices l l }1 . ..\−quad representingce˘0 Thed real .. cycles0 axis−se˘1 $ v =ce˘0 0a $ i s represented0 −se˘1 by the ray coming through \quad ( 0 , 0 , 1 , 0 ) \quad and a matrix from .. open parenthesis 2 period 6 closing parenthesis .. are invariant under the similarity with elements of K comma .. thus they are \ centerlinei . e .{ $ the\ l e real f t (\ linebegin is {SLarray2(R)}{− invariantcc } s \ .breve 2 .{e} A{ direct1 } & 0 calculation\\ 0 & − ins GiNaC\breve{e} { 1 }\end{ array }\ right ) indeed [ 4 6 , § 3 . 2 . 1 ] shows that matrices representing cycles . $K hyphen\quad orbitsFor period any \quad $\ l e f t (\ begin { array }{ cc } a & b \breve{e} { 0 }\\ − c \breve{e} { 0 } & d \end{ arrayfrom}\ right ( 2 .) 6 )\ in areSL invariant{ 2 } under( the R similarity )$ wehave: with elements} of K, thus they are It iindeed s surprising on the first glance that the C sub sigma-breve to the power of s i s defined through a Clifford algebra lscript-C open parenthesis sigma-breve closing parenthesis \ [ \ l e f t (\ begin { array }{ cc } a & b \breve{e} { 0 }\\ − c \breve{e} { 0 } & d \end{ array }\ right ) \ l e f t (\ begin { array }{ cc } s with an arbitrary sign of 2 e-breve sub 1 sub periodK ....− orbits However. a moment of reflections reveals that transformation open parenthesis 3 period\breve 5{ closinge} { parenthesis1 } & 0 \\ 0 & − s \breve{e} { 1 }\end{ array }\ right ) \ l e f t (\ begin { array }{ cc } d & − b \breve{e} { 0 }\\ c \breve{e} { 0 } &s a \end{ array }\ right ) = \ l e f t (\ begin { array }{ cc } s depends only fromIt i s the surprising sign of 0 e-breve on the to first the glancepower of that 2 but the doesC notσ˘ i involves defined any through quadratic a open Clifford parenthesis algebra or higher closing parenthesis t\breve ermslscript of{ breve-ee} −{C sub1(˘σ)} 1& period 0 \\ 0 & − s \breve{e} { 1 }\end{ array }\ right ), \ ] Remarkwith .... an 3 arbitrary period 5 period sign ....of Such2e˘1 . However a variety of a choicesmoment i s ofa consequence reflections of reveals the usage that of SLtransformation sub 2 open parenthesis ( 3 R closing parenthesis endash. a 5 smaller ) \noindent i . e . \quad the real2 line is $ SL { 2 } (R) − $ invariant . groupdepends of symmetries only from in comparison the sign of t o0 thee˘ but all doesM o-dieresis not involve bius maps any of quadratic R to the power ( or higher of 2 period ) t erms .... The of SLe˘1. sub 2 open parenthesis R 2 . \quad A \quad d i r e c t \quad calculation \quad in \quad GiNaC \quad [ 4 \quad 6 , \quad \S \quad 3 . 2 . 1 ] \quad shows \quad that \quad matrices \quad representing \quad c y c l e s closingRemark parenthesis group 3 . fixes 5 . the Such a variety of choices i s a consequence of the usage of SL2(R) – a realsmaller line and consequently a decomposition of vectors into quotedblleft real quotedblright open parenthesis e sub 0 closing parenthesis and \ hspace ∗{\ f i l l }from \quad ( 2 . 6 ) \quad are invariant under the2 similarity with elements of $ K , $ quotedblleftgroup imaginary of symmetries quotedblright in comparison open parenthesis t o the e sub all 1 M closingo¨ bius parenthesis maps of partsR . The SL2(R) group fixes \quadi s obviousthethus period they .. are This indeed p ermits t o assign an arbitrary value t o the square of the quotedblleft imaginary unit quotedblright e sub 1 e sub 0 periodreal line and consequently a decomposition of vectors into “ real ” (e0) and “ imaginary ” (e1) \ [KGeometricparts− invariants io sr b obvious i t s defined . .\ ] below This comma p ermits e period t o g assignperiod orthogonalities an arbitrary in value Sections t o 4 the period square 1 and of 4 period the “ 3 comma demonstrate quotedblleftimaginary awareness unit ” quotedblrighte1e0. Geometric .. of the invariants real line invariance defined below in one , way e . or g another. orthogonalities period .. We in will Sections call this the boundary effect4 . in 1 the and upper 4 . 3 half , demonstrate hyphen plane geometry “ awareness period ” .. The of the famous real question line invariance .. o n .. h in e a one .. r way i n g or d r another .. u .. quoteright m s .. s .. ha p e ..\ hspace has .∗{\ Wef i will l l } It call i this s surprising the boundary on effect the firstin the glance upper half that - plane the geometry $ C ˆ{ s .} {\ Thebreve famous{\sigma}}$ i s defined through a Clifford algebra $ l s c r i p t −C( \breve{\sigma} ) $ 0 a sistquestion er : o n h e a r i n g d r u m s s ha p e has a sist er : Can we see slash feel theCan boundary we see from / insidefeel the a domain boundary ? from inside a domain ? \noindentRemarksRemarks 3with period 3 . an 1 1 3 3 arbitrary comma , 4 ... 1 4 0period sign and 1 of 0 4 and . $ 20 .. 2 4 provide period{\breve 20 hints provide{e} for{ hints positive1 }} for positive{ answers. }$ answers\ . h f i periodl l However a moment of reflections reveals that transformation ( 3 . 5 ) s To encompassTo all asp encompass ects from allopen asp parenthesis ects from 3 period ( 3 . 3 closing ) we think parenthesis a cycle we thinkCσ˘ defined a cycle byC sub a quadruple sigma-breve to the power of s defined by\noindent a quadruple(k, l, n,depends m open) as parenthesis only from k comma the l sign comma of n comma $ 0 m{\ closingbreve parenthesis{e}}ˆ{ as2 }$ but does not involve any quadratic ( or higher ) t erms of $ \anbrevean quotedblleft “{e imageless} { imageless1 ” . obj}$ quotedblright ect which obj have ect distinct which have implementations distinct implementations ( a circle open , parenthesisa parabola a or circle a comma a parabola or a hyperbolahyperbola closing parenthesis ) σ \noindentin thein corresponding theRemark corresponding\ spaceh f i l lR space to3 the . 5 powerR . .\ h ofThese f sigmai l l Such period implementations a .. variety These implementations of may choices lookvery may i s look different a consequencevery different , see Fig comma of . the see Fig usage period of .. 6 $ open SL { 2 } parenthesis( R6 ( a) a ) closing$ ,−− but parenthesisa still s m have a l l e rcomma some properties in common . For example , but still have some properties in common period .. For example comma \noindent group of symmetries in comparison t o the all M $ \ddot{o} $ bius maps of $Rˆ{ 2 } . $ \ h f i l l The $ SL { 2 } ( R )$ group fixes the

\noindent real line and consequently a decomposition of vectors into ‘‘ real ’’ $ ( e { 0 } ) $ and ‘‘ imaginary ’’ $ ( e { 1 } ) $ parts i s obvious . \quad This p ermits t o assign an arbitrary value t o the square of the ‘‘ imaginary unit ’’ $ e { 1 } e { 0 } . $ Geometric invariants defined below , e . g . orthogonalities in Sections 4 . 1 and 4 . 3 , demonstrate ‘‘ awareness ’’ \quad of the real line invariance in one way or another . \quad We will call this the boundary effect in the upper half − plane geometry . \quad The famous question \quad o n \quad h e a \quad r i n g d r \quad u \quad $ ’ { m }$ s \quad s \quad ha p e \quad has a s i s t er :

\ centerline {Can we see / feel the boundary from inside a domain $ ? $ }

\noindent Remarks 3 . 1 3 , \quad 4 . 1 0 and \quad 4 . 20 provide hints for positive answers .

\ hspace ∗{\ f i l l }To encompass all asp ects from ( 3 . 3 ) we think a cycle $ C ˆ{ s } {\breve{\sigma}}$ definedbyaquadruple $( k , l , n , m )$ as

\noindent an ‘‘ imageless ’’ obj ect which have distinct implementations ( a circle , a parabola or a hyperbola )

\noindent in the corresponding space $ R ˆ{\sigma } . $ \quad These implementations may look very different , see Fig . \quad 6 ( a ) , but still have some properties in common . \quad For example , 14 .... V period V period Kisil \noindenthline14 14 \ h f i l l V . V . K i s i l V . V . Kisil bullet .. All implementations have the same vertical axis of symmetries period \ [ bullet\ r u l e..{ Intersections3em}{0.4 pt with}\ the] real axis open parenthesis if exist closing parenthesis coincide comma see r sub 1 and r sub 2 for the left cycle in Fig period 6 open parenthesis a closing parenthesis period bullet .. Centres of• circleAll c implementations sub e and corresp onding have the hyperbolas same vertical c sub h are axis mirror of symmetries reflections of . each other \ centerlinein the real• axis{Intersections$ with\ bullet the parabolic with$ \quad the centre realAll b e axis in implementations the ( if middle exist point ) coincide period have , see ther same1 and verticalr2 for the leftaxis cycle of symmetries in . } LemmaFig 2. period6 ( a ) 5 . gives another example of similarities b etween different implementations of the same \ hspacecycles∗{\ defined• fCentres i l byl } the$ of\ equationbullet circle opence$and parenthesis\quad correspIntersections onding 2 period hyperbolas 6 closing with parenthesisc theh are real mirror period axis reflections ( if exist of each ) other coincide , see $ r { 1 }$ andFinallyin $r the comma{ real2 we} axis$ may for with restate the the the parabolicleft Proposition cycle centre in3 period Fig b e 3in . as 6the an ( middleintertwining a ) . point property . period CorollaryLemma .... 2 3 period. 5 gives 6 period another .... Any example .... implementation of similarities .... of b cycles etween .... different shown .... onimplementations .... open parenthesis of the 3 period 3 closing parenthesis ....$ by\ bullet ....same the ....cycles$ dotted\quad defined ....Centres arrow by forthe ....of equation any circle ( 2 $ . c 6 ){ . e }$ and corresp onding hyperbolas $ c { h }$ are mirror reflections of each other incombination the realFinally of axis sigma with ,comma we may the sigma-breve restate parabolic the andProposition s centre intertwines b e two 3 in . actions 3 the as an middle of intertwining SL sub point 2 open property .parenthesis . R closing parenthesis : .... by matrix conjugationCorollary open parenthesis 3 . 6 3 . periodAny 4 closing implementation parenthesis .... and of cycles shown on ( 3 . 3 ) by the \noindentM dieresis-odottedLemma bius transformations 2 . 5 gives open another parenthesis example 2 arrowperiod of 3 for similarities closing parenthesis b etweenperiod different implementations any of the same cyclesRemarkcombination defined 3 period 7 ofby period theσ, σ˘ .. equationand A similars intertwines representation ( 2 . 6 two ) of . actionscircles by of 2 timesSL2 2(R complex): by matrix matrices conjugation which intertwines( 3 . M dieresis-o4 ) bius transformations and matrix conjugations was used recently by A period A period Kirillov openand square bracket 3 .. 1 closing square\ centerlineM bracketo¨ bius{ inFinally the transformations , we may restate( 2 . 3 ) . the Proposition 3 . 3 as an intertwining property . } studyRemark of the Apollonian 3 . 7 . gasket periodA similar .. Kirillov representation quoteright s of matrix circles realisation by 2 × ..2 opencomplex square matrices bracket 3 which 1 closing square bracket .. of a cycle\noindent hasintertwines an attractiveC o r o l l M a r yo¨ \biush f i ltransformations l 3 . 6 . \ h f iand l l Any matrix\ h f conjugations i l l implementation was used\ recentlyh f i l l o byf c A y c . l e A s .\ h f i l l shown \ h f i l l on \ h f i l l ( 3 . 3 ) \ h f i l l by \ h f i l l the \ h f i l l dotted \ h f i l l arrow f o r \ h f i l l any quotedblleftKirillov self [ 3 hyphen 1 ] in adjoint the quotedblright form : \noindentC substudy sigma-brevecombination of the Apollonian to the power of gasket $of s\sigma = Row . 1 Kirillov m, l breve-e\breve ’ s sub matrix{\ 0sigma plus realisation sn} e-breve$ and sub [ 1 $ 3 Row 1 s ] $ 2 minus of intertwines a cycle l breve-e has sub antwo 0 minus actions sn e-breve of sub $ SL 1 k { 2 } ( Rattractive ) : $ “ self\ h -f i adjoint l l by matrix ” form : conjugation ( 3 . 4 ) \ h f i l l and . open parenthesis in notations of this paper closing parenthesis period .. open parenthesis 3 period 6 closing parenthesis Note that the matrix inverses to .... openm parenthesis le˘0 + 3 periodsne˘1 6 closing parenthesis i s intertwined with the FSCc presentation open parenthesis Cσ˘ = ( in notations of this paper ) . ( 3 . 6 ) 3\noindent period 2 closingM $ parenthesis\ddot{o} ....$ by− the biusle˘0 − transformationssne˘1 k ( 2 . 3 ) . matrixNote .. Rowthat 1 the 0 1 matrix Row 2 1 inverse 0 . period to ( 3 . 6 ) i s intertwined with the FSCc presentation ( 3 . 2 ) \noindent3 periodby the 2Remark .. First invariants 3 . 7 . of\ cyclesquad A similar representation of circles by $ 2 \times 2 $ complex matrices which intertwines M $ \ddot{o} $0 1bius transformations and matrix conjugations was used recently by A . A . Kirillov [ 3 \quad 1 ] in the Usingmatrix implementations from. Definition 3 period 1 and relation open parenthesis 3 period 4 closing parenthesis we can derive some invariants of 1 0 \noindentcycles3 . open 2study parenthesis First of invariants the under Apollonian the M of dieresis-o cycles gasket bius transformations . \quad Kirillov closing parenthesis’ s matrix from realisation well hyphen known\quad invariants[ 3 1 of] \ matrixquad openof a cycle has an attractive parenthesis‘ ‘ sUsing e l f under− implementationsadjoint simi hyphen ’’ form from : Definition 3 . 1 and relation ( 3 . 4 ) we can derive some invariants laritiesof closing parenthesis period .. First we use trace to define an invariant inner product in the space of cycles period \ hspaceDefinitioncycles∗{\ 3f ( i period under l l } $ 8 the C period ˆ M{ s ....o¨ }bius Inner{\ transformationsbreve sigma-breve{\sigma hyphen}} ) from product= well\ l of e f two- t known(\ cyclesbegin invariants i{ sarray given by}{ cc of the} matrix tracem & of ( their underl product\breve as{e matrices} { 0 : } + sn Equation:simi\breve - open{e} parenthesis{ 1 }\\ 3 − periodl 7 closing\breve parenthesis{e} { 0 ..} angbracketleft − sn C\ subbreve sigma-breve{e} { 1 to} the& power k \end of comma{ array to}\ theright power) of ( s $ tilde-Cin notationslarities sub sigma-breve ) . of this First to the paper we power use ) traceof . s right\quad to defineangbracket( 3 an . 6invariant = ) tr open parenthesisinner product C sub inbreve-sigma the space to of the cycles power . of s to the power of C-tilde sub breve-sigmaDefinition to the 3 power . 8 . of sInner closing parenthesisσ˘− product periodof two cycles i s given by the trace of their product \noindentTheas above matricesNote definition that: i s verythe similar matrix t o inverse an inner product to \ h defined f i l l ( in 3 operator . 6 ) algebras i s intertwined .. open square with bracket the 1 closing FSCc square presentation bracket period ( 3 . 2 ) \ h f i l l by the This i s not a coincidence : .. cycles act on points of R to the power of sigma by inversions comma see Subsection .. 4 period 2 comma and this\noindent matrix \quad $\ l e f t (\ begin { array }{ cc } 0 & 1 \\ 1 & 0 \end{ array }\ right ) . $ s ˜s sC˜ s action i s linearised by FSCc comma thus cycleshC canσ˘, C beσ˘ i viewed= tr(Cσ˘ asσ˘ linear). operators as well period (3.7) \noindentGeo metrical3 .interpretation 2 \quad First of the inner invariants product will of b cycles e given in Corollary 5 period 8 period The above definition i s very similar t o an inner product defined in operator algebras [ An obvious but interesting observation i s that for matrices representing cyclesσ we obtain the \noindentsecond1 ] classical. ThisUsing invariant i s implementations not a open coincidence parenthesis from : determinant cycles Definition act closing on 3 pointsparenthesis . 1 and of R under relationby similarities inversions ( 3 open . , see 4 parenthesis ) Subsection we can 3 periodderive 4 closing some parenthesis invariants of from the4. first 2 , open and parenthesis this action trace i s closing linearised parenthesis by FSCc as follows , thus : cycles can be viewed as linear operators \noindentEquation:as wellcycles open . parenthesis ( under 3 period the M 8 closing $ \ddot parenthesis{o} $ .. bius angbracketleft transformations C sub sigma-breve ) from to wellthe power− known of comma invariants to the power of of matrix s C ( under simi − sub breve-sigmaGeo tometrical the powerinterpretation of s right angbracket of the = inner minus product 2 determinant willC b sube given sigma-breve in Corollary to the power 5 . 8 of . period to the power of s \noindentThe explicitAnl obvious a expression r i t i e s but ) for . interesting the\quad determinantFirst observation i we s : use i trace s that to for define matrices an representing invariant cycles inner we product obtain in the space of cycles . Equation:the second open parenthesis classical invariant 3 period 9 ( closing determinant parenthesis ) under .. determinant similarities C sub ( sigma-breve3 . 4 ) from to the the first power ( of trace s = l to the power of 2 minus breve-sigma\noindent) as follows sDefinition to the :power of 3 2 . n to8 the . \ powerh f i l l ofInner 2 minus mk $ \ periodbreve{\sigma} − $ product of two cycles i s given by the trace of their product as matrices : We recall that the same cycle is defined by any matrix lambda C sub sigma-breve to the power of comma to the power of s lambda in R sub \ begin { a l i g n ∗} plus comma thus the determinant comma s s s hC , C i = −2 det C . (3.8) \ langleeven being MC dieresis-o ˆ{ s } {\ biusbreve hyphen{\ invariantsigma} commaˆ{σ˘ , σ˘ is}}\ usefultilde onlyσ˘ in{C the} ˆ identities{ s } {\ of thebreve sort{\ detsigma C sub}}\ sigma-breverangle to the= power t r of s = ( 0 C ˆ{ s } {\breve{\sigma}}ˆ{\ tilde {C}}ˆ{ s } {\breve{\sigma}} ). \ tag ∗{$ ( 3 . 7 ) $} periodThe .. Note explicit also expression for the determinant i s : \endthat{ a tr l ig open n ∗} parenthesis C sub sigma-breve to the power of s closing parenthesis = 0 for any matrix of the form open parenthesis 3 period 2 closing parenthesis period Since it may b e convenient to have a predefined The above definition i s very similar ts o an2 inner2 2 product defined in operator algebras \quad [ 1 ] . representative of a cycle out of the ray of equivalentdet Cσ˘ = FSCcl − matricesσs˘ n − mk. we introduce the following (3.9) Thisnormalisation i s not period a coincidence : \quad cycles act on points of $ R ˆ{\sigma }$ by inversions , see Subsection \quad 4 . 2 , and this action i s linearised by FSCc , thus cycles can be vieweds as linear operators as well . We recall that the same cycle is defined by any matrix λCσ˘, λ ∈ R+, thus the determinant , s even being M o¨ bius - invariant , is useful only in the identities of the sort det Cσ˘ = 0. Note \ centerline {Geo metricals interpretation of the inner product will b e given in Corollary 5 . 8 . } also that tr (Cσ˘ ) = 0 for any matrix of the form ( 3 . 2 ) . Since it may b e convenient to have a predefined representative of a cycle out of the ray of equivalent FSCc matrices we introduce An obviousthe following but interesting observation i s that for matrices representing cycles we obtain the secondnormalisation classical . invariant ( determinant ) under similarities ( 3 . 4 ) from the first ( trace ) as follows :

\ begin { a l i g n ∗} \ langle C ˆ{ s } {\breve{\sigma} ˆ{ , }} C ˆ{ s } {\breve{\sigma}}\rangle = − 2 \det C ˆ{ s } {\breve{\sigma} ˆ{ . }}\ tag ∗{$ ( 3 . 8 ) $} \end{ a l i g n ∗}

\noindent The explicit expression for the determinant i s :

\ begin { a l i g n ∗} \det C ˆ{ s } {\breve{\sigma}} = l ˆ{ 2 } − \breve{\sigma} s ˆ{ 2 } n ˆ{ 2 } − mk . \ tag ∗{$ ( 3 . 9 ) $} \end{ a l i g n ∗}

We recall that the same cycle is defined by any matrix $ \lambda C ˆ{ s } {\breve{\sigma} ˆ{ , }}\lambda \ in R { + } , $ thus the determinant , even being M $ \ddot{o} $ bius − invariant , is useful only in the identities of the sort det $ C ˆ{ s } {\breve{\sigma}} = 0 . $ \quad Note a l s o that t r $ ( C ˆ{ s } {\breve{\sigma}} ) = 0 $ for any matrix of the form ( 3 . 2 ) . Since it may b e convenient to have a predefined representative of a cycle out of the ray of equivalent FSCc matrices we introduce the following

\noindent normalisation . Erlangen Program at Large hyphen 1 : .... Geometry of Invariants .... 1 5 \noindenthlineErlangenErlangen Program Program at Large at - Large 1 :− 1 : \ h f i l l GeometryGeometry of of Invariants Invariants \ h f i l l 1 5 1 5 Figure 7 period .. Different sigma hyphen implementations of the same sigma-breve hyphen zero hyphen radius cycles and corresponding fo ci\ [ period\ r u l e {3em}{0.4 pt }\ ] Definition .. 3 period 9 period .. A FSCc matrix representing a cycle i s said to be k hyphen normalised if it s .. open parenthesis 2 comma 1 closingFigure parenthesis 7 . hyphenDifferent σ− implementations of the sameσ ˘− zero - radius cycles and corresponding fo ci . \ centerlineelementDefinition i s 1{ Figure and it i s 3 7 det . . 9 hyphen\quad . normalisedAD i FSCc f f e r e n ifmatrix t its $ determinant\ representingsigma i− s equal$ a cycle implementations 1 period i s said to be ofk− thenormalised same $ \breve{\sigma} − $ zeroEach−if normalisationitradius s ( 2 cycles , 1 has ) - its and element .. own corresponding advantages i s 1 and it : .. i fo s element det ci - . normalised ..} open parenthesis if its determinant 1 comma 1 closing i s equal parenthesis 1 . .. of k hyphen normalised matrix immeEach hyphen normalisation has its own advantages : element ( 1 , 1 ) of k− normalised \noindentdiatelymatrix .. tD ell imme e f us i n .. i t the-i o n diately ..\ centrequad ..3 oft . ell the 9 us .. . cycle\quad the commaA FSCc centre .. meanwhile matrix of the.. representing det hyphen cycle normalisation , a meanwhile cycle .. i i s s .. said preserved det - to ..be by ..$ matrix k − $ normalised if it s \quad ( 2 , 1 ) − elementconjugationnormalisation i s with 1 and SL sub iti s 2 iopen preserveds det parenthesis− normalised by R closing matrix parenthesis if itsconjugation determinant .. element with .. openSL i2 s( parenthesisR equal) element 1 which . is ( importantwhich in view of Proposition 3 periodis 3 important closing parenthesis in view period of Proposition .. The lat er 3 . 3 ) . The lat er Eachnormalisationnormalisation normalisation is used is comma hasused its , for for example\ examplequad own comma , in advantages in[ 3 open 1 ] square : \ bracketquad 3element 1 closing\ squarequad bracket( 1 , 1 ) \quad o f $ k − $ normalised matrix imme − d iTaking a t e l yTaking into\quad account intot e its account l l invariance us \ itsquad invariance it ithe s not\ surprisingquad it i sc not e n that t r surprising e the\quad determinanto that f the the of\ a determinantquad cycle entersc y c l e the of , a\quad cycle entersmeanwhile \quad det − normalisation \quad i s \quad preserved \quad by \quad matrix conjugationfollowingthe following Definition with 3 Definition period $ SL 1{ 0 .. 32 of .} the 1( 0 focus R of and the ) the $ focus invariant\quad and zeroelement the hyphen invariant\ radiusquad zero cycles( which - radius from is Definition cycles important from 3 period in 1view 2 period of Proposition 3 . 3 ) . \quad The l a t er Definition 3 . 1 2 . Definition 3 period 10 period ring-sigma hyphen Focuss of a cycle C sub breve-sigmaσ to the power of s i s the point in R to the power of sigma \noindentEquation:Definitionnormalisation open parenthesis3.10. ˚σ− 3 periodisFocus used 10of closing , a for cycle parenthesis exampleCσ˘ i s the .. , f insub point [ring-sigma 3 in 1R ] = parenleftbigg l divided by k sub comma minus determinant C sub sigma-ring to the power of s divided by 2 nk parenrightbigg or explicitly f sub ring-sigma = parenleftbigg l divided by k sub comma mk Taking into account its invariance it i s not surprising that the determinant of a cycle enters the minus l to the power of 2 plus sigma-ringl ndet to theCs power of 2 divided by 2l nkmk parenrightbigg− l2 + ˚σn2 period following Definition 3 . 1 0 \quad ˚σof the focus and the invariant zero − radius cycles from Definition 3 . 1 2 . We also use e hyphen focusf comma˚σ = ( p− hyphen focus) orexplicitly comma h hyphenf˚σ = ( focus and ring-sigma). hyphen focus comma(3.10) in line with Convention 2 k , 2nk k , 2nk period 1 t o take into account \noindent Definition $3 . 10 . \ mathring {\sigma} − $ Focus of a cycle $Cˆ{ s } {\breve{\sigma}}$ of theWe typ also e of use C-lscripte - focus open parenthesis, p - focus sigma-ring, h - focus closingand parenthesis˚σ− focus period, in line with Convention 2 . 1 t i s the point in $Rˆ{\sigma }$ Focalo take length into of a account cycle is n dividedof the typ by 2 e k of subC period− lscript(˚σ). Remark 3 period 1 1 period .... Note thatFocal focus of length C sub sigma-breveof a cycle tois then power of s is independent of the sign of s period .... Geometrical \ begin { a l i g n ∗} 2k . meaning of s Remark 3 . 1 1 . Note that focus of Cσ˘ is independent of the sign of s. Geometrical f focus{\ imathring s as follows{\ periodsigma ..}} If a cycle= is ( realised\ f r a c in{ thel }{ parabolick } { space, } R to− the \ f power r a c {\ ofdet p h hyphenC ˆ{ focuss } comma{\ mathring p hyphen{\ focussigma comma}}}{ 2 nk } meaning) or of explicitly f {\ mathring {\sigma}} = ( \ f r a c { l }{ k } { , }\ f r a c { mk − l ˆ{ 2 } .. e hyphenfocus focus i s as are follows . If a cycle is realised in the parabolic space Rp h - focus , p - focus , + correspondingly\ mathring {\ geometricalsigma} focusn ˆ{ of2 the}}{ parabola2 nk comma} its). vertex\ tag and∗{ the$ point ( on3 directrix . 10 nearest ) $} \end{ea l - i g focus n ∗} are correspondingly geometrical focus of the parabola , its vertex and the point on t odirectrix the vertexnearest comma see t Figo the period vertex 6 open , see parenthesis Fig . 6 ( b b closing ) . parenthesis Thus the period traditional .. Thus focus the traditional i s h - focus focus i s h hyphen focus in our notationsin ourperiod notations . \noindentWe may describWe a l e s oa finer use structure e − f o c of u sthe , cycle p − spacef o c u through s , h − invariantf o c u s subclasses and $ of\ mathring them period{\sigma} − $ focus , in line with Convention 2 . 1 t o take into account of theWe typ may e of describ $C− el s a cfiner r i p t structure ( \ mathring of the cycle{\sigma space} through) . invariant $ subclasses of them Two. such Two families such families are zero hyphen are zero radius - radius and selfand hyphenself adjoint- adjoint cyclescycles which which are naturally are naturally appearing appearing from expressions open parenthesis 3 period 8 closing parenthesis and open parenthesis 3 period 7 closing parenthesis corresp ondingly period \ centerlinefrom expressions{ Focal length ( 3 . 8 of ) and a cycle ( 3 . 7 $ ) correspis \ f r aondingly c { n }{ . 2 k } { . }$ } Definition .... 3 period 1 2 period sigma-breve hyphen Zero hyphen radius cycles are defined bys the condition det open parenthesis C sub Definition 3.12. σ˘− Zero - radius cycles are defined by the condition det (Cσ˘ ) = 0, i . e . breve-sigma to the power of s closing parenthesis = 0 comma i period e period are ex hyphen \noindentare exRemark - 3 . 1 1 . \ h f i l l Note that focus of $ C ˆ{ s } {\breve{\sigma}}$ is independent of the sign of plicitlyplicitly given given by matrices by matrices $ sEquation: . $ open\ h f parenthesis i l l Geometrical 3 period 1 meaning 1 closing parenthesis of ..Row 1 y minus y to the power of 2 Row 2 1 minus y . = 1 divided by 2 Row 1 y y Row 2 1 1 . Row 1 1 minus y Row 2 1 minus y . = Row 1 e-breve sub 0 u plus breve-e sub 1 v u to the power of 2 minus breve-sigma v to \noindent focus i s as2 follows  . \quadIf  a cycle is realised in the2 parabolic2  space $ R ˆ{ p }$ h − f o c u s , p − f o c u s , \quad e − f o c u s are the power of 2 Row 2 1 minusy −y e-breve sub1 0 uy minus y e-breve1 − suby 1 v . commae˘0u +e ˘1v u − σv˘ correspondingly geometrical= focus of the parabola= , its vertex and the point, on(3 directrix.11) nearest where y = e-breve sub1 0 u− plusy breve-e2 sub1 1 v 1 period ..1 We− denotey such a sigma-breve1 −e˘ hyphen0u − e˘1 zerov hyphen radius cycle by Z sub breve-sigma tot the o power the vertex of s open , parenthesis see Fig y . closing 6 ( b parenthesis ) . \quad periodThus the traditional focus i s h − focus in our notations . s Geometricallywhere y = sigma-brevee ˘0u +e ˘1v. hyphenWe denote zero hyphen such aradiusσ˘− zero cycles - are radius sigma cycle hyphen by implementedZσ˘ (y). by Q from Definition 3 period 1 rather diffe Wehyphen may describGeometrically e a finerσ˘− structurezero - radius of thecycles cycle are σ space− implemented through byinvariantQ from Definition subclasses 3 . of 1 them . Tworentlyrather such comma families diffe see - Fig period are zero .. 7 period− radius .. Some and notable self rules− areadjoint : cycles which are naturally appearing from expressionsrently , see ( Fig3 . . 8 ) 7 and . ( Some 3 . notable7 ) corresp rules are ondingly : .

\noindent D e f i n i t i o n \ h f i l l $ 3 . 1 2 . \breve{\sigma} − $ Zero − radius cycles are defined by the condition det $ ( C ˆ{ s } {\breve{\sigma}} ) = 0 ,$ i.e.areex −

\noindent plicitly given by matrices

\ begin { a l i g n ∗} \ l e f t (\ begin { array }{ cc } y & − y ˆ{ 2 }\\ 1 & − y \end{ array }\ right ) = \ f r a c { 1 }{ 2 }\ l e f t (\ begin { array }{ cc } y & y \\ 1 & 1 \end{ array }\ right ) \ l e f t (\ begin { array }{ cc } 1 & − y \\ 1 & − y \end{ array }\ right ) = \ l e f t (\ begin { array }{ cc }\breve{e} { 0 } u + \breve{e} { 1 } v & u ˆ{ 2 } − \breve{\sigma} v ˆ{ 2 }\\ 1 & − \breve{e} { 0 } u − \breve{e} { 1 } v \end{ array }\ right ), \ tag ∗{$ ( 3 . 1 1 ) $} \end{ a l i g n ∗}

\noindent where $ y = \breve{e} { 0 } u + \breve{e} { 1 } v . $ \quad We denote such a $ \breve{\sigma} − $ zero − radius cycle by $ Z ˆ{ s } {\breve{\sigma}} ( y ) . $

\ hspace ∗{\ f i l l } Geometrically $ \breve{\sigma} − $ zero − radius cycles are $ \sigma − $ implemented by $ Q $ from Definition 3 . 1 rather diffe −

\noindent rently , see Fig . \quad 7 . \quad Some notable rules are : 1 6 .... V period V period Kisil \noindenthline1 6 1 6 \ h f i l l V . V . K i s i l V . V . Kisil open parenthesis sigma sigma-breve = 1 closing parenthesis Implementations are zero hyphen radius cycles in the standard sense : the point ue\ [ sub\ r u0 l minus e {3em ve}{ sub0.4 1 pt }\ ] in elliptic case and the light cone with the centre at ue sub 0 plus ve sub 1 in hyperbolic space open square bracket 1 6 closing square bracket period (σσ˘ = 1) Implementations are zero - radius cycles in the standard sense : the point ue0 − ve1 \ hspaceopenin parenthesis∗{\ ellipticf i l l } case sigma$ ( and = 0\ thesigma closinglight parenthesis cone\brevewith{\ .. Implementationssigma the centre} = at areue 10 parabolas+ )ve $1 in Implementations hyperbolic with focal length space v slash [ are 1 6 2 ] zero and . the− realradius axis passing cycles in the standard sense : the point $ uethrough{ 0 the(}σ sigma-breve= − 0) veImplementations hyphen{ 1 }$ focus period are parabolas In other words with comma focal for length breve-sigmav/2 and = minus the real 1 focus axis at openpassing parenthesis u comma v closing parenthesisthrough open parenthesis the σ˘− thefocus real . axis In other is directrix words closing , for parenthesisσ˘ = −1 focus comma at (u, v)( the real axis is directrix \ centerlinefor), sigma-breve{ in = elliptic 0 focus at case.. open and parenthesis the light u comma cone v slash with 2 the closing centre parenthesis at open $ ue parenthesis{ 0 } the+ real ve axis{ passes1 }$ through in hyperbolic the space [ 1 6 ] . } vertex closingfor parenthesisσ˘ = 0 focus comma at for(u, v/breve-sigma2) ( the = real 1 focus axis at passes through the vertex ) , for σ˘ = 1 focus at \ hspaceopen parenthesis∗{\(u,f0) i l l (} the u$ comma ( real\ axis 0sigma closing passes parenthesis= through 0 open ) the $ parenthesis focus\quad ) .Implementations the Such real axis parabolas passes through are as well parabolas the have focus closing “ with zero parenthesis focal length period .. $ Such v parabolas/ 2- $ radius as and well ” have the .. real quotedblleft axis passing zero hyphen radius quotedblright for a suitable parabolic metricfor comma a suitable see Lemma parabolic 5 period metric 7 period , see Lemma 5 . 7 . \ hspaceopen parenthesis∗{\(˘σ =f 0) i lσ l−} through sigma-breveImplementations the = 0 closing $ \ arebreve parenthesis corresponding{\sigma sigma} conic hyphen − $ sections Implementationsfocus . which In other t are ouch corresponding words the real , foraxis conic . $ sections\breve which{\sigma t ouch} the= real −axis period1$Remark focusat 3 . 1 3 $( . The u above , v “ t ouching ) ($ ” property thereal of axis zero - is radius directrix cycles for ) , σ˘ = 0 is an Remarkexample 3 period of 1 3 period .... The above quotedblleft t ouching quotedblright property of zero hyphen radius cycles for sigma-breve = 0 is\ hspace an exampleboundary∗{\ f of i l l effect} f o r inside $ \breve the domain{\sigma mentioned} = 0 in $ Remark f o c u s 3 at . 5\quad . It i s$( not surprising u , aft v er / all 2 ) ($ the real axis passes throughthe vertex ) , for $ \boundarybrevesince{\ effectsigma inside} = the domain 1 $ mentioned f o c u s at in Remark 3 period 5 period .... It i s not surprising aft er all since SLSL sub2( 2R open) action parenthesis on the R upper closing half parenthesis - plane action may on b the e considered upper half hyphen as an plane extension may b of e considered it s action as anon extension of it s action on the\ hspacethe∗{\ f i l l } $( u , 0 ) ($ the real axis passes through the focus ) . \quad Such parabolas as well have \quad ‘ ‘ zero − r a d i u s ’ ’ realreal axis axis period . \ centerlinesigma-breve{ hyphenforσ˘− Zero a Zero suitable - hyphen radius radiusparabolic cycles cycles are significant are metric significant , since see since Lemma they they are are 5 completely. 7 . } determined determined by their by their centres and thuscentres .... quotedblleft and encode quotedblright points into the .... quotedblleft cycle language quotedblright period .... The following result states that\ centerline thisthus encoding “{ encode$ ( ”\breve points{\ intosigma the} “= cycle 0 language ) \sigma ” . The− following$ Implementations result states that are this corresponding conic sections which t ouch the real axis . } i s Mencoding o-dieresis bius invariant as well period \noindenti s M Remarko¨ bius invariant 3 . 1 3 as . well\ h f i . l l The above ‘‘ t ouching ’’ property of zero − radius cycles for $ \breve{\sigma} Lemma .... 3 period 14 period .... The .... conjugate−1 gs to the power of minus 1 Z sub sigma-breve tos the power of s open parenthesis y =closing 0Lemma parenthesis $ is an g 3 example .... . 14 of a . breve-sigma ofThe conjugate hyphen zero hypheng Zσ˘ (y radius)g of ....a cycleσ˘− zero Z sub - radiussigma-breve cycle to theZ powerσ˘ (y) with of s open parenthesis y closing g ∈ SL2(R) is parenthesis .... with g in SL sub 2 opens parenthesis R closing parenthesis .... is \noindenta σ˘−boundaryzero - radius effect cycle insideZσ˘ (g · y the) with domain centre mentioned at g · y – the in M Remarko¨ bius 3 transform . 5 . \ h off i l the l It centre i s not surprising aft er all since a sigma-breves hyphen zero hyphen radius cycle Z sub breve-sigma to the power of s open parenthesis g times y closing parenthesis with centreof at g timesZσ˘ (y). y endash the M o-dieresis bius transform of the centre of Z sub sigma-breve to the power of s open parenthesis y closing parenthesis\noindentProof period$ . SL { 2 } ( R ) $ actionThis on may the be upper calculated half in− Gplane i NaC may[ 4 b 6 e , § considered2.7].  as an extension of it s action on the ProofAnother period .... important This may classb e calculated of cycles in isG igiven NaC open by next square definition bracket 4 based 6 comma on S the 2 period invariant 7 closing inner square pro bracket period blacksquare \noindentAnother- importantr e a l a x class i s . of cycles is given by next definition based on the invariant inner pro hyphen ductduct open ( parenthesis 3 . 7 ) and 3 period the invariance 7 closing parenthesis of the real and line the . invariance of the real line period \ hspaceDefinition∗{\ f i l l } $ \breve 3 . 1{\ 5sigma . }Self − -$ adjoint Zero cycle− radiusfor σ˘ cycles6= 0 are defined significant by the conditionsince they are completely determined by their centres and Definitions ....s 3 period 1 5 period .... Self hyphen adjoint cycle for sigma-breve equal-negationslash 0 are defined by the condition Re

\ hspace ∗{\ f i l l }( p ) \quad vertical lines , which are also \quad ‘‘ parabolic circles ’’ \quad [ 72 ] , i . e . are givenby $ \ parallel x − y \ parallel = r ˆ{ 2 }$ in the

\ centerline { parabolic metric defined below in ( 5 . 3 ) . }

\noindent Lemma 3 . 16 . \quad S e l f − adjoint cycles form a family , which is invariant under the M $ \ddot{o} $ bius transfor − mations .

\noindent Proof . \ h f i l l The proof i s either geometrically obvious from the transformations describ ed in Sec −

\noindent t ion 2 . 2 , or follows analytically from the action describ ed in Proposition $ 3 . 3 . \ blacksquare $

\noindent Remark 3 . 1 7 . \ h f i l l Geometric obj ects , which are invariant under infinitesimal action of $ SL { 2 } ( R ) , $

\noindent were studied recently in papers [ 5 3 , \quad 5 2 ] .

\noindent 4 \quad Joint \quad invariants : \quad orthogonality \quad and \quad i n v e r s i o n s

\noindent 4 . 1 \quad Invariant orthogonality type conditions

\noindent We already use the matrix invariants of a single cycle in Definitions 3 . 1 0 , 3 . 1 2 and 3 . 1 5 . \ h f i l l Now

\noindent we will consider j oint invariants of several cycles . \ h f i l l Obviously , the relation tr $ ( C ˆ{ s } {\breve{\sigma}}ˆ{\ tilde {C}}ˆ{ s } {\breve{\sigma}} ) = 0$ between

\noindent two cycles i s invariant under M $ \ddot{o} $ bius transforms and characterises the mutual disp osition of two

\noindent c y c l e s $ C ˆ{ s } {\breve{\sigma}}$ and $ \ tilde {C} ˆ{ s } {\breve{\sigma} ˆ{ . }}$ \quad More generally the relations

\ hspace ∗{\ f i l l } t r $ ( p ( C ˆ{ s ( 1 ) } {\breve{\sigma}} , . . . , C ˆ{ s ( n ) } {\breve{\sigma}} ) ) = 0 $ \quad or \quad det $ ( p ( C ˆ{ s ( 1 ) } {\breve{\sigma}} , . . . , C ˆ{ s ( n ) } {\breve{\sigma}} ) ) = 0 ( 4 . 1 ) $ Erlangen Program at Large hyphen 1 : .... Geometry of Invariants .... 1 7 \noindentLineErlangen 1 hlineErlangen Line Program 2 variables Program at Largeto the at power - Large 1 : of b− etween1 : \ nh sigma-breve f i l l GeometryGeometry from of open of Invariants Invariants parenthesis n closing\ h f i l parenthesisl 1 7 1 7 to cycles prime x comma period period period comma x to the power of C to the power of s to the power of open parenthesis sub i s to the power of 1 closing parenthesis to\ [ \ thebegin power{ a of l i gcomma n e d }\ periodr u l e period{3em}{ M0.4 period pt }\\ o-dieresis comma C sub breve-sigma to the power of s bius sub invariant to the power of open parenthesisv a r i a b l n e closings ˆ{ b parenthesis etween to the n power}\ ofbreve based{\ onsigma if p to} theˆ{ power(of n a open ) } parenthesis{ c y c l e s polynomial{\prime x to the{ x power} of,... open parenthesis , x }ˆ{ C }ˆ{ s }}ˆ{ ( }ˆ{ 1 ) } { i s }ˆ{ , } .. { M } . {\( ddot{o}} (n), C ˆ{ s } {\breve{\sigma}}{ bius }ˆ{ ( 1 closing parenthesisbetween comman (n) period( period1), periods comma(n) xbased to the powera of open parenthesis( nx closing1), parenthesis...,x )of p closing parenthesis sub i s(i) n ) variables} { i n v a r i a nσ˘ tcycles}ˆC{ baseds is ..M}.o,¨on Cσ{˘ biusi f invariant} p ˆ{ onifa }p (polynomial( polynomialx(1),...,x(n) p)is{ homogeneousx ˆ{ ( 1 )ninnon} ,...,− commutingeveryx . Non− to the power of open parenthesis0x,...,x x to the power of open parenthesis sub homogeneous to the power of 1 closing parenthesis sub comma to the powerx ˆ{ of( period n period ) }} periodp{ ) comma}ˆ{ ( x to the x ˆ power{ ( }} of open{ i parenthesis s }ˆ{ n1 closing ) parenthesis{ , }} { closinghomogeneous parenthesis} ofˆ{ n. in non . hyphen . commuting , x ˆ{ ( o¨ everyn ) xhomogeneous to} the) power o f of} polynomials openn{ in parenthesis} non will i closing also− create parenthesiscommuting M periodbius{ every Noninvariants hyphen} x if ˆ{ we( substitute i ) } cycles. ’ Non det - − \end{ a l i g n e d }\ ] homogeneousnorma - polynomials lised matrices will alsoonly create . MLet dieresis-o us consider bius invariants some lower if we order substitute realisations cycles quoteright of ( 4 . det 1 ) hyphen . norma hyphen s ˜s lisedDefinition matrices only 4 period . 1 . .. LetTwo us consider cycles someCσ˘ and lowerC orderσ˘ are realisationsσ˘− orthogonal of open parenthesisif the real 4 period part 1 of closing their parenthesis period \noindentDefinitioninner prohomogeneous 4 period - 1 period polynomials .... Two cycles will C sub also sigma-breve create to M the power$ \ddot of s{o ....} and$ tilde-Cbius invariants sub sigma-breve if to we the substitute power of s .... cycles are ’ det − norma − breve-sigmalisedduct matrices hyphen ( 3 . 7 orthogonal ) only vanishes . \ ifquad : the realLet part us .... consider of their inner some pro lowerhyphen order realisations of ( 4 . 1 ) . duct open parenthesis 3 period 7 closing parenthesis vanishes : \noindent Definition 4 . 1 . \ h f i l l Two cycles $ C ˆ{ s } {\breve{\sigma}}$ \ h f i l l and $ \ tilde {C} ˆ{ s } {\breve{\sigma}}$ Equation: open parenthesis 4 periods 2s closing parenthesis .. Re angbracketlefts s C˜ s Cs sub sigma-breve to the power of comma to the power of s

Equation: open parenthesis 4 period 4 closing2 parenthesis2 .. k open parenthesis u to the power of 2 k minus sigma v to the power of 2 closing parenthesis\ hspace ∗{\ minusf i l l 2}Note angbracketleft that the open orthogonalityk parenthesis(u k − σv ) l− comma2h( identityl, n) n, ( closingu, σv˘ ) (i parenthesis+ 4m .= 3 0, a comma) is linear open parenthesis for coefficients u(4 comma.4) sigma-breve of one v cycle closing if the other parenthesis right angbracket plus m = 0 comma i . e .σ− implementation of Cs is passing through the point (u, σv˘ ), which σ˘− centre of \noindenti period ecycle period sigma i s fixed hyphen . implementation\quad σThus˘ of we C obtain sub sigma-breve several to thesimple power conclusions of s is passing .through the point open parenthesis u Zs(u, v). comma breve-sigmaσ v closing parenthesis comma which sigma-breve hyphen centre of Z sub sigma to the power of s open parenthesis u comma \noindent CorollaryThe important 4 . 4 consequence . of the above observations is the possibility to extrapolate v closingresults parenthesis period Thefrom important zero -consequence radius cycles of the to above the entire observations space is . the possibility to extrapolate results \ centerlinefrom zero hyphen{1 . radius\quad cyclesA $ to\ thebreve entire{\sigma space period} − $ s e l f − orthogonal cycle is $ \breve{\sigma} − $ zero − radius one ( 3 . 1 1 ) . } \ hspace ∗{\ f i l l }2 . \quad For $ \breve{\sigma} = \pm 1 $ \quad there is no non − trivial cycle orthogonal to \quad all o ther non − trivial cycles . \quad For

\ centerline { $ \breve{\sigma} = 0 $ only the real axis $ v = 0 $ is orthogonal to all other non − trivial cycles . }

\ hspace ∗{\ f i l l }3 . \quad For $ \breve{\sigma} = \pm 1 $ \quad any \quad c y c l e \quad i s \quad uniquely \quad d e f i n e d \quad by the family \quad o f c y c l e s \quad orthogonal to \quad i t , \quad i . e .

\ hspace ∗{\ f i l l } $ ( C ˆ{ s ˆ{\bot }} {\breve{\sigma}} ) ˆ{\bot } = \{ C ˆ{ s } {\breve{\sigma}} \} . $ \quad For $ \breve{\sigma} = 0 $ \quad the s e t $ ( C ˆ{ s ˆ{\bot }} {\breve{\sigma}} ) ˆ{\bot }$ \quad consists of all cycles which have the same

\ centerline { r o o t s as $ C ˆ{ s } {\breve{\sigma} ˆ{ , }}$ see middle co lumn of pictures in Fig . 8 . }

\ centerline {We can visualise the orthogonality with a zero − radius cycle as follow : }

\noindent Lemma 4 . 5 . \quad A c y c l e $ C ˆ{ s } {\breve{\sigma}}$ i s $ \breve{\sigma} − $ orthogonal to $ \sigma − $ zero − radius cycle $ Z ˆ{ s } {\sigma } ( u , v ) $ \quad i f

\ begin { a l i g n ∗} k ( u ˆ{ 2 } k − \sigma v ˆ{ 2 } ) − 2 \ langle ( l , n ) , ( u , \breve{\sigma} v ) \rangle + m = 0 , \ tag ∗{$ ( 4 . 4 ) $} \end{ a l i g n ∗}

\noindent i . e $ . \sigma − $ implementation of $ C ˆ{ s } {\breve{\sigma}}$ is passing through the point $ ( u , \breve{\sigma} v ) , $ which $ \breve{\sigma} − $ centre of $Zˆ{ s } {\sigma } ( u , v ) . $

\ hspace ∗{\ f i l l }The important consequence of the above observations is the possibility to extrapolate results

\noindent from zero − radius cycles to the entire space . 1 8 .... V period V period Kisil \noindenthline1 8 1 8 \ h f i l l V . V . K i s i l V . V . Kisil Figure 8 period .. Orthogonality of the first kind in nine combinations period Each picture presents two groups open parenthesis green \ [ and\ r u blue l e {3em closing}{0.4 parenthesis pt }\ ] of cycles which are orthogonal to the red cycle C sub sigma-breve to the power of period to the power of s .. Point b belongs to C sub breve-sigma to the power of s and the family of Figure 8 . Orthogonality of the first kind in nine combinations . Each picture presents two groups ( green blue cycles passing through b is orthogonal to C sub sigma-breves to the power of periods to the power of s .. They all also intersect in the \noindentand blueFigure ) of cycles 8 . which\quad areOrthogonality orthogonal to the red of cycle theC firstσ˘. Point kindb belongs in nine to C combinationsσ˘ and the family of. blueEach picture presents two groups ( green point d which is the s andcycles blue passing ) of cycles through b whichis orthogonal are orthogonal to Cσ˘. They to all the also intersect red cycle in the $ point C ˆd{ whichs } {\ is thebreve{\sigma} ˆ{ . }}$ \quad Point inverse of b in C subs sigma-breve to the power of period to the power of s .. Any orthogonality is reduced to the usual orthogonality with a $b$inverse belongs of b in toCσ˘. $CˆAny{ orthogonalitys } {\breve is reduced{\sigma to the}} usual$ and orthogonality the family with a of new ( “ ghost ” ) cycle ( new open parenthesis quotedblleft ghost quotedblright closing parenthesiss cycle blueopenshown cycles parenthesis by the passing dashedshown by line through the ) , dashed which $ may line bclosing$ or may is parenthesis not orthogonal coincide comma with toC whichσ˘. $ CFor may ˆ{ anys or point} may{\ nota breveon coincide the{\ “ ghostsigma with ” C} cycle subˆ{ sigma-breve. }}$ \quad to theThey power all also intersect in the point of$ periodd $the towhich orthogonality the power is the of is s reduced.. For any to point the lo a cal on notion the quotedblleft in the terms ghost of tangent quotedblright lines at the intersection point . Consequently such a point a is always the inverse of itself . cycle the orthogonality is reduced to the lo cal notion3 in the3 terms of tangent lines at the intersection \noindentpointProposition periodinverse Consequently 4 of .6 $b$ such . a point inLet a $Cˆ is alwaysT {: Ps the→} inverse{\P breve ofis itself an{\ orthogonalitysigma period} ˆ{ . preserving}}$ \quad mapAny of orthogonality the is reduced to the usual orthogonality with a new ( ‘‘ ghost ’’ ) cycle s ˜s s ˜s ( showncycles by space the , dashed i . line e . )hC ,σ˘, whichCσ˘ i = 0 may⇔ or hTC mayσ˘, T C notσ˘ i = coincide 0. Then with for σ $6= C 0 ˆthere{ s } is{\ a mapbreve{\sigma} ˆ{ . }}$ Propositionσ 4 periodσ 6 period .. Let T : P to the power of 3 right arrow P to the power of 3 .. is an orthogonality preserving map of the cycles\quad spaceTσFor: R comma any→ R point .., i periodsuch $ that e a period $Q on the ‘‘ ghost ’’ cycle the orthogonality is reduced to the lo cal notion in the terms of tangent lines at the intersection angbracketleft C sub sigma-breve to the power ofintertwinesT comma to the andT power: of s tilde-C sub sigma-breve to the power of s right angbracket = 0 Leftrightarrow angbracketleft TC sub breve-sigma to the power of commaσ to the power of s T C-tilde sub breve-sigma to the power of s right angbracket\noindent =point 0 period . .. Consequently Then for sigma suchequal-negationslash a pointQTσ = T $ 0Q. a there $(4.5) is is a mapalways T sub the sigma inverse : R to theof power itself of sigma . right arrow R to the power of sigma comma .. such that Q \noindentProofProposition . If T preserves 4 . 6 . \ thequad orthogonalityLet $ T : ( i P . eˆ{ .3 }\ therightarrow inner productP ˆ ({ 33 .} 7$ ) \quad is an orthogonality preserving map of the cycles space , \quad i . e . Line 1 intertwines T and T sub sigma : Line 2 QT sub sigma = TQ period open parenthesis 4 periods 5 closing parenthesis $ \andlangle consequentlyC ˆ{ s } the{\breve determinant{\sigma} fromˆ{ (, 3}}\ . 8 )tilde ) then{C} ˆ by{ s the} image{\breveTZ{\σ˘ (sigmau, v) }}\of a rangle = 0 \Leftrightarrow Proof period .. If T preservess the orthogonality .. open parenthesis i periods e period .. the inner product .. open parenthesis 3 period 7 zero - radius cycle Z (u, v) i s again a zero - radius cycle Z (u1, v1) and we can define Tσ by closing\ langle parenthesisTC ˆ{ .. ands } ..{\ consequentlyσ˘breve{\sigma the } ˆ{ , }} T \ tilde {Cσ˘} ˆ{ s } {\breve{\sigma}}\rangle = 0 . $ the\quad identityThenT f oσ r:(u, $ v)\7→sigma(u1, v1). \ne 0 $ there is amap $T {\sigma } : R ˆ{\sigma }\rightarrow determinant from open parenthesis 3 period 8 closing parenthesis closing parenthesis .. then by the images TZ sub sigma-breve to the power R ˆ{\sigmaTo prove} , the $ intertwining\quad such property that $Q$ ( 4 . 5 ) we need t o show that if a cycle Cσ˘ passes of s open parenthesis u comma v closing parenthesiss .. of a zero hyphen radius cycle Z sub breve-sigma to the power of s open parenthesis u through (u, v) then the image TCσ˘ passes through Tσ(u, v). However for σ 6= 0 this i s a commaconsequence v closing parenthesis of .. i s again \ [ \abegin zero hyphen{ a l i g n radius e d } intertwines cycle Z sub sigma-breve T and to the Tpower{\ ofsigma s open} parenthesis: \\ u sub 1 comma v sub 1 closing parenthesis and we can defineQT T{\ subsigma sigma by} the= identity TQ T . sub sigma ( 4 : open . parenthesis 5 ) \ uend comma{ a l i v g nclosing e d }\ ] parenthesis mapsto-arrowright open parenthesis u sub 1 comma v sub 1 closing parenthesis period To prove the intertwining property open parenthesis 4 period 5 closing parenthesis we need t o show that if a cycle C sub sigma-breve to the\noindent power of sProof passes through . \quad If $ T $ preserves the orthogonality \quad ( i . e . \quad the inner product \quad ( 3 . 7 ) \quad and \quad consequently the determinantopen parenthesis from u comma ( 3 . v 8 closing ) ) parenthesis\quad then then by the the image image TC sub $ sigma-breve TZ ˆ{ s } to{\ thebreve power{\ ofsigma s passes}} through( T u sub , sigma v open ) $ parenthesis\quad o f u a comma zero − v closingradius parenthesis cycle period $ Z ˆ ..{ Howevers } {\ forbreve sigma{\ negationslash-equalsigma}} ( u0 this , i s a v consequence ) $ \ ofquad i s again a zero − radius cycle $ Z ˆ{ s } {\breve{\sigma}} ( u { 1 } , v { 1 } ) $ and we can define $ T {\sigma }$ by the identity $T {\sigma } : ( u , v ) \mapsto ( u { 1 } , v { 1 } ) . $

To prove the intertwining property ( 4 . 5 ) we need t o show that if a cycle $ C ˆ{ s } {\breve{\sigma}}$ passes through $( u , v )$ thentheimage $TCˆ{ s } {\breve{\sigma}}$ passes through $ T {\sigma } ( u , v ) . $ \quad However for $ \sigma \not= 0 $ this i s a consequence of Erlangen Program at Large hyphen 1 : .... Geometry of Invariants .... 1 9 \noindenthlineErlangenErlangen Program Program at Large at - Large 1 :− 1 : \ h f i l l GeometryGeometry of of Invariants Invariants \ h f i l l 1 9 1 9 the T hyphen invariance of orthogonality and the expression of the point hyphen t o hyphen cycle incidence through the \ [ orthogonality\ r u l e {3em}{ from0.4 Lemma pt }\ ] 4 period 5 period blacksquare Corollary 4 period 7 period .. Let T sub i : P to the power of 3 right arrow P to the power of 3 comma i = 1 comma 2 .. are two orthogonality preservingthe mapsT − ofinvariance the cycles of orthogonality and the expression of the point - t o - cycle incidence \noindentthroughthe the $ orthogonality T − $ invariance from Lemma of orthogonality4.5.  and the expression of the point − t o − cycle incidence through the space period .. If they coincide on the subspace3 of3 sigma-breve hyphen zero hyphen radius cycles comma breve-sigma negationslash-equal 0 commaorthogonalityCorollary then they are 4from identical . 7 Lemma . in Let $ 4Ti : P .→ 5P , i = . 1, 2\ blacksquareare two orthogonality $ preserving maps of the thecycles whole P to space the power . of If 3 they period coincide on the subspace of σ˘− zero - radius cycles , σ˘ 6= 0, then \noindentthey areCorollary identical 4 in . 7 . \quad Let $ T { i } : P ˆ{ 3 }\rightarrow P ˆ{ 3 } , i = 1 Remark 4 period 83 period .. Note comma that the orthogonality i s reduced t o lo cal notion in t erms of tangent lines ,t o 2the cycles $ whole\ inquad theirPare intersection. two orthogonality points .. only for preserving sigma sigma-breve maps = of 1 comma the cycles .. i period e period .. this happens only in NW and SE spacecornersRemark . of\quad Fig period 4If . 8 they8 . period coincideNote .. In other , that on cases the the the orthogonality subspace lo cal condition of i s can $ reduced\ bebreve formulated{\ t osigma lo in cal} t erm notion − of$ quotedblleft in zero t erms− ghostradius of quotedblright cycles cycle$ , \breve{\sigma} \notdefined=tangent 0b elow lines, period $ then t o cycles they in are their identical intersection in points only for σσ˘ = 1, i . e . this Wehappens denote by only chi open in NW parenthesis and SE sigma corners closing of parenthesis Fig . 8 . the In Heaviside other casesfunction the : lo cal condition can be \noindentEquation:formulatedthe open whole parenthesis in t erm $ of P 4 period “ ˆ{ ghost3 6} closing ” cycle. $ parenthesis defined ..b chielow open . parenthesis t closing parenthesis = braceleftbigg minus 1 sub comma to the power of 1 comma t to the powerWe denote of t greater by χ equal(σ) the lessHeaviside 0 sub period function to the power: of 0 comma \noindentPropositionRemark 4 period 4 9 . period 8 . ....\quad Let cyclesNote C , sub that sigma-breve the orthogonality and tilde-C sub i sigma-breve s reduced be t breve-sigma o lo cal hyphen notion orthogonal in t erms period of .... tangent lines Fort theiro cycles sigma hyphen in their implementations intersection we define points \quad only f o r $ \sigma \breve{\sigma} = 1 , $ \quad i . e . \quad this happens only in NW and SE χ(t) = { −11, tt ≥ <00, (4.6) cornersthe ghost of cycle Fig C-circumflex . 8 . \quad sub breve-sigmaIn other .. cases by the following the, lo two cal conditions. condition : can be formulated in t erm of ‘‘ ghost ’’ cycle defined1 period chib elow open parenthesis . sigma closing parenthesis hyphen centre of C-circumflex sub breve-sigma .. coincides with sigma-breve hyphen Proposition 4 . 9 . Let cycles Cσ˘ and C˜σ˘ be σ˘− orthogonal . For their σ− centre ofimplementations C sub breve-sigma we semicolon define \ centerline2 period ..{ determinantWe denote of by C-circumflex $ \ chi sub( sigma\sigma to the power) $ of the 1 is equal Heaviside to determinant function of C : sub} sigma to the power of chi open the ghost cycle Cˆ by the following two conditions : parenthesis breve-sigma closingσ˘ parenthesis period 1. χ(σ)− centre of Cˆ coincides with σ˘− centre of C ; \ beginThen{ :a l i g n ∗} σ˘ σ˘ ˆ1 χ(˘σ) \ chi1 period( C-circumflex t )2 sub =. sigma\{ determinant .. co − incides1 ˆ ofwith{ 1 Cσ sub ,is} equalsigma{ , if to} sigma determinantt ˆ breve-sigma{ t }\ ofgeq ={ 1C<σ semicolon}. 0 ˆ{ 0 , } { . }\ tag ∗{$ ( 4 .2 period 6 ) C-circumflex $} sub sigma has common roots open parenthesis real or imaginary closing parenthesis with C sub sigma semicolon \end3 period{ a l i g n ..∗} in the sigma hyphen implementation the tangent line to C-tilde sub breve-sigma at points of its intersections with the ghost T hen : cycle C-circumflex sub sigma .. are passing the sigma hyphen centre of circumflex-C sub sigma to the power of period \noindentProof periodProposition .... The calculations 4 . 9 are. \ doneh f i l inl Let G i NaC cycles comma $ see C open{\ squarebreve{\ bracketsigma 4}} 6 comma$ and S 3 $ period\ tilde 3 period{C} 4{\ closingbreve square{\sigma}}$ 1. Cˆ co incides with C if σσ˘ = 1; bracketbe $ period\breve ....{\ Forsigma illustration} − $ see Fig orthogonal periodσ 8 comma . \ h where f i l l Forσ t h e i r $ \sigma − $ implementations we define 2. Cˆ has common roots ( real or imaginary ) with C ; the ghost cycle i s shown by theσ black dashed line period blacksquare σ σ− C˜ \noindentConsiderationthe3 . of ghost the in ghost the cycle cycle doesimplementation $ \hat present{C} the{\ orthogonality thebreve tangent{\sigma in linethe}} local to$ t\ ermsσ˘quadat however pointsby the of it its following intersections two conditions : hideswith the the symmetry ghost of this relation period Cˆ σ− Cˆ . \ centerlineRemark 4 period{ $ 1 10 period . Elliptic\ chicycle and( hyperbolicσ \sigmaare passing ghost) cycles the− $ are symmetric ccentre e n t r e of o inf the $σ real\hat line{C} comma{\breve the parabolic{\sigma}}$ \quad coincides with $ \ghostbreveProof cycle{\sigma has. itThe} s centre calculations − $ on itcentre comma are of see done Fig $C in periodG{\ i NaC8breve period,{\ see ....sigma [This 4 6}} is , an§3 illustration .; 3 $ . 4} ] . to For the illustration boundary effect see from RemarkFig . 3 8 period , where 5 period . \ centerline4 periodthe ghost 2 ..{2 Inversions cycle . \quad i s in shown cyclesdeterminant by the black of $dashed\hat{ lineC} ˆ{ 1 } {\sigma }$ is equal to determinant of $ C ˆ{\ chi ( Definition\breveConsideration{\ 3 periodsigma} 1 associates of) the} {\ ghost asigma 2 times cycle} 2 does hyphen. $ present} matrix the to orthogonalityany cycle period in .... the Similarly local t ermst o SL however sub 2 open parenthesis R closing parenthesisit action open parenthesis 2 period 3 closing parenthesis we can \ beginconsiderhides{ a l ai g the fraction n ∗} symmetry hyphen linearof this transformation relation . on R to the power of sigma defined by such a matrix : ThenEquation:Remark : open 4 parenthesis . 10 . 4Elliptic period 7 and closing hyperbolic parenthesis ghost .. C sub cycles sigma are to the symmetric power of s in : ue the sub real 0 plus line ve , sub the 1 arrowright-mapsto C sub sigma\end{ toparabolica l the i g n power∗} of s open parenthesis ue sub 0 plus ve sub 1 closing parenthesis = open parenthesis le sub 0 plus ne sub 1 closing parenthesis open parenthesisghost cycle ue hassub 0 it plus s centre ve sub on 1 closing it , see parenthesis Fig . 8 . plus This m divided is an illustration by k open parenthesis to the boundary ue sub 0 plus effect ve sub 1 closing parenthesis minus\ centerline openfrom parenthesis{ $ 1 l e . sub 0\hat plus{ neC} sub{\ 1 closingsigma parenthesis}$ \quad subco comma incides with $ C {\sigma }$ i f $ \sigma \breve{\sigma} =where 1Remark C ; sub $ sigma 3} . 5 to . the power of s is as usual open parenthesis 3 period 2 closing parenthesis C sub4 . sigma 2 to Inversions the power of sin = cycles Row 1 le sub 0 plus ne sub 1 m Row 2 k minus le sub 0 minus ne sub 1 . period \ centerlineDefinition{ $ 3 2 . 1 associates . \hat{ aC}2 ×{\2− sigmamatrix} to$ any has cycle common . Similarly roots ( t realo SL2 or(R) imaginaryaction ( 2 . ) 3 with ) $ C {\sigma } ; $ we} can consider a fraction - linear transformation on Rσ defined by such a matrix : \ hspace ∗{\ f i l l }3 . \quad in the $ \sigma − $ implementation the tangent line to $ \ tilde {C} {\breve{\sigma}}$ at points of its intersections with the ghost (le + ne )(ue + ve ) + m Cs : ue + ve 7→ Cs(ue + ve ) = 0 1 0 1 (4.7) σ 0 1 σ 0 1 k(ue + ve ) − (le + ne ) \ centerline { c y c l e $ \hat{C} {\sigma }$ \quad are0 passing1 the0 $1 \, sigma − $ c e n t r e o f $ \hat{C} {\sigma ˆ{ . }}$ } s where Cσ is as usual ( 3 . 2 ) \noindent Proof . \ h f i l l The calculations are done in G i NaC , see [ 4 6 , \S 3 . 3 . 4 ] . \ h f i l l For illustration see Fig . 8 , where  le + ne m  Cs = 0 1 . σ k −le − ne \noindent the ghost cycle i s shown by the black dashed0 1 line $ . \ blacksquare $

\ hspace ∗{\ f i l l } Consideration of the ghost cycle does present the orthogonality in the local t erms however it

\noindent hides the symmetry of this relation .

\noindent Remark 4 . 10 . Elliptic and hyperbolic ghost cycles are symmetric in the real line , the parabolic

\noindent ghost cycle has it s centre on it , see Fig . 8 . \ h f i l l This is an illustration to the boundary effect from

\noindent Remark 3 . 5 .

\noindent 4 . 2 \quad Inversions in cycles

\noindent Definition 3 . 1 associates a $ 2 \times 2 − $ matrix to any cycle . \ h f i l l Similarly t o $ SL { 2 } ( R )$ action(2.3)wecan

\noindent consider a fraction − linear transformation on $ R ˆ{\sigma }$ defined by such a matrix :

\ begin { a l i g n ∗} C ˆ{ s } {\sigma } : ue { 0 } + ve { 1 }\mapsto C ˆ{ s } {\sigma } ( ue { 0 } + ve { 1 } ) = \ f r a c { ( l e { 0 } + ne { 1 } ) ( ue { 0 } + ve { 1 } ) + m }{ k ( ue { 0 } + ve { 1 } ) − ( l e { 0 } + ne { 1 } ) } { , }\ tag ∗{$ ( 4 . 7 ) $} \end{ a l i g n ∗}

\noindent where $ C ˆ{ s } {\sigma }$ is as usual (3 . 2)

\ [ C ˆ{ s } {\sigma } = \ l e f t (\ begin { array }{ cc } l e { 0 } + ne { 1 } & m \\ k & − l e { 0 } − ne { 1 }\end{ array }\ right ). \ ] 20 .... V period V period Kisil \noindenthline20 20 \ h f i l l V . V . K i s i l V . V . Kisil Another natural action of cycles in the matrix form i s given by the conjugation on other cycles : \ [ Line\ r u 1l e C{3em sub}{ sigma-breve0.4 pt }\ to] the power of s : tilde-C sub sigma-breve to the power of s mapsto-arrowright C sub breve-sigma to the power of s toAnother the power natural of C-tilde action sub breve-sigma of cycles in to the the powermatrix of form s C sub i s sigma-brevegiven by the to conjugation the power of period on other to the cycles power of s open parenthesis 4 period: 8 closing parenthesis Line 2 Note that equivalent C t o to the power of sigma-breve to the power of s sub expressions to the power of C\noindent sub breve-sigmaAnother to the natural power of actions to the power of cycles of = minus in the determinant matrix sub form C sub i s sigma-breve given by to the the power conjugation of s to the on power other of C-tilde cycles : sub breve-sigma to the power of s C sub sigma-breve to the power of s to the power of open parenthesis C sub breve-sigma to the power of s \ [ \ begin { a l i g n e d } C ˆ{ s } {\breve{\sigma}} : \ tilde {C} ˆ{ s } {\breve{\sigma}}\mapstos ˜s C ˆ{sC˜s s }s {\breve{\sigma}}ˆ{\ tilde {C}}ˆ{ s } {\breve{\sigma}} closing parenthesis I comma sub minus to the power of where 1 for to the power of I is determinant the C sub breve-sigmaC toσ˘ : theC powerσ˘ 7→ Cσ of˘ sσ˘ Cσ˘. (4.8) C ˆ{ s } {\breve{\sigmas s } ˆ{ . }} (s 4 . 8 ) \\ identity equal-negationslashσ˘ 0C sinceσ˘ matrix= − cyclesdet(Cσ˘ sub)I, formwhere to theI power of period to the power of Thus. a theThus proj definition ective space open NotethatequivalentC 1for isdettheC s identity6=0sincematrixcycles atheprojdefinition ( 4.8) isThere Note that { e q u i vto aexpressions l e n t } C { Cts C˜ s Cs o− }ˆ{\breve{\sigmaσ˘ } ˆ{ s }}ˆ{ C ˆ{ sform} {\breve{\sigma}}}ectivespace{ expressions. }ˆ{ = parenthesis period 4 period 8 closing parenthesisσ˘ i sσ˘ Thereσ˘ − \det }ˆ{ ( C ˆ{ s } {\breve{\sigma}} )I, } { C ˆ{ s } {\breve{\sigma}}ˆ{\ tilde {C}}ˆ{ s } {\breve{\sigma}} i s ai connection s a connection between between two actions two open actions parenthesis ( 4 . 4 7 period ) and 7 ( closing 4 . 8 parenthesis ) of cycles and , open which parenthesis is similar 4 period t o 8 closing parenthesis of Ccycles ˆ{ commas } {\ whichbreve is similar{\sigma t o}}} SL subˆ{ 2where open parenthesis} { − } R1 closing f o parenthesisr ˆ{ I } actioni s {\det } the { C } {\breve{\sigma} ˆ{ s }} i d e n t i t y {\ne } SL2(R) action 0in since Lemma 3 matrix period 14{ periodc y c l e s }ˆ{ . } { form }ˆ{ Thus } a the { pr oj } d e f i n i t i o n { ective space } ( { . } 4 .in Lemma 8 ) 3 . i 14 . s { There }\end{ a l i g n e d }\ ] Lemma 4 period 1 1 period .. Let det C subs sigma-breve to the power of s negationslash-equal 0 comma then : Lemma 4 . 1 1 . Let det Cσ˘ 6= 0, then : 1 period ..1 The . conjugation The conjugation open parenthesis( 4 . 84 period) preserves 8 closing the parenthesis orthogonality preserves relation the orthogonality( 4 . 2 ) . relation open parenthesis 4 period 2 closing parenthesis period ˜ s2Z s1 s2 ˜s1 \noindent2 .i s a The connection image betweenCσ σ (u, two v)Cσ actionsof a (σ 4− .zero 7 ) - radius and ( 4 . cycle 8 ) ofZσ cyclesunder , the which is similar t o 2 periodconjugation .. The .. image( 4 C . sub 8 ) sigmais to the power of s to the power of 2 Z-tilde sub sigma to the power of s sub 1 open parenthesis u comma v$ closing SL { parenthesis2 } ( C R sub sigma ) $ to athe c t i o power n of s to the power of 2 .. of a sigma hyphen zero hyphen radius .. cycle tilde-Z sub sigma to the ˜s1 0 0 0 0 power of s to the powera σ of− 1zero .. under - radius the .. cycle conjugationZσ ( ..u open, v ), parenthesiswhere ( 4u period, v ) 8is closing calculated parenthesis by the .. linear is - \noindentfractionalin Lemma trans - 3 . 14 . a sigma hyphen zero hyphen radius cycle0 Z-tilde0 subs1s2 sigma to the power of s to the power ofs1s 12 open parenthesis u to the power of prime comma v to the powerformation of prime closing(4.7) parenthesis (u , v ) = commaCσ (u, .. v where) associated open parenthesis to the ucycle to theC powerσ . of prime comma v to the power of prime 3closing\noindent . parenthesis BothLemma formulae .. is 4 calculated .( 1 4 1 . . 7 by\ )quad theand linearLet( hyphen 4 det . 8 ) fractional$ Cdefine ˆ{ s trans the} {\ hyphen samebreve transformation{\sigma}}\ ofnot the= point 0 space , $ . then : formationProof open . parenthesisThe first part 4 period i s obvious 7 closing, parenthesis the second open i s parenthesis calculated u in toG the i NaCpower[ of 4 prime 6 , § comma3 . 2 . v 3 to ] . the power of prime closing parenthesis\ centerlineThe = last C{1 sub part . sigma\quad to theThe power conjugation of s to the power ( 4 .of 8 1 to ) the preserves power of s the 2 open orthogonality parenthesis u comma relation v closing ( parenthesis 4 . 2 ) .. . associated} to the cyclefollows C sub from sigma the to first the twopower and of s Proposition to the power of4. 16. to the power of s 2 period \ hspace3 period∗{\There ..f Both i l l are}2 formulae at . \quad least openThe twoparenthesis\ naturalquad 4image period ways t 7 $ o closing C define ˆ{ parenthesiss inversions} {\sigma .. inand cycles} openˆ{ 2 parenthesis .\ Onetilde of 4{Z period them}}ˆ{ 8use closings { 1 parenthesis}} {\sigma .. define} ( uthe , v ) C ˆ{ s } {\sigma }ˆ{ 2 }$ \quad o f a $ \sigma − $ zero − r a d i u s \quad c y c l e $ \ tilde {Z} ˆ{ s } {\sigma }ˆ{ 1 }$ the same transformation of the point space period 00 \quadProoforthogonalityunder period the .... The\quad condition firstconjugation part , i another s obvious define\ commaquad them( the 4 second . as 8 “ ) ireflections s\quad calculatedi s in incycles G i NaC open square bracket 4 .... 6 comma S 3 period 2 period 3 closing square bracket period .... The last part . \ hspacefollowsDefinition∗{\ fromf i the l l } firsta 4 .$ two 1\ 2sigma and . Proposition− $ 4 zero period− 6radius period blacksquare cycle $ \ tilde {Z} ˆ{ s } {\sigma }ˆ{ 1 } ( u ˆ{\prime } , v ˆ{\prime } ) , $ \quad s where $ ( u ˆ{\prime } , v0 ˆ{\prime } ) $ \quad is calculated by the linear − fractional trans − There are1 . at ..Inversion least two naturalin a cycle ways tC oσ definesends inversions a point inp cyclesto the period second .. One point of themp of use intersection the of all orthogonalitycycles condition comma another define them as quotedblleft reflections in Case 1 quotedblright Case 2 period \ centerline { formation $( 4 . 7 )s ( uˆ{\prime } , v ˆ{\prime } ) = C ˆ{ s } {\sigma }ˆ{ 1 ˆ{ s } Definition 4 period 1 2 period orthogonal t o Cσ and passing through p. 2 } ( u , v ) $ \quad associateds to the−1 cycle $ C ˆ{ s } {\sigma s}ˆ{ 1 ˆ{ s } 2 } . $ } 1 period2 .. . InversionReflection in a cyclein aC cycle sub sigmaCσ toi s the given power by ofM s sendsRM awhere point pM to thesends second the point cycle p toCσ theinto power the of prime of intersection of all cycleshorizontal \ centerlineorthogonal t{ o3 C . sub\quad sigmaBoth toaxis the formulae and powerR ofis s theand ( 4 passingmirror . 7 ) through reflection\quad pand period in that( 4 .axis 8 ). \quad define the same transformation of the point space . } 2 periodWe .. Reflection are in going a cycle C to sub sigma see to that the power inversions of s i s given are by M givento the power by of minus ( 4 . 17 RM ) where and M sends the cycle C sub sigma\noindent toreflections the powerProof of. sare into\ h f the i expressed l l horizontalThe first part i s obvious , the second i s calculated in G i NaC [ 4 \ h f i l l 6 , \S 3 . 2 . 3 ] . \ h f i l l The last part axisthrough and R is ( the 4 .mirror 8 ) , reflection thus they in arethat essentially axis period the same in light of Lemma 4 . 1 1 . \noindentWeRemark .. are ..follows going 4 . .. 1 to from 3 .. . see the ..Here that first i.. s inversions a two simple and .. example are Proposition .. given where .. by usage .. $ open 4 of parenthesis . complex 6 4 . period ( dual\ blacksquare 7 or closing double parenthesis ) $ .. and .. reflections .. are ..numbers expressed \ hspacethroughi s∗{\ weaker openf i l parenthesis l } thenThere Clifford are 4 period at algebras\ 8quad closing ,least parenthesis see two Remark comma natural 2 thus. 2 ways . they areA t reflection o essentially define of the inversions a same cycle in in light the in of axis cyclesLemma 4 . period\quad 1 1One period of them use the Remarkv = 4 period 0 1 3 period .... Here i s a simple example where usage of complex .... open parenthesis dual or double closing parenthesis  e˘ 0  ....\noindent numbersi s representedorthogonality by the conjugationcondition , ( 4another . 8 ) with define the corresp them as onding ‘‘ reflections matrix 1 in $\ l. eThe f t . c y c l e s \ begin { a l i g n e d } & 0 −e˘ ’’ i\\ s weaker then Clifford algebras comma .... see Remark 2 period 2 period .... A reflection of a cycle in1 the axis v = 0 &.same\end{ a l i g n e d }\ right . $ i s representedtransformation by the in conjugation t erm of open complex parenthesis numbers 4 period should 8 closing involve parenthesis a complex with the conjugation corresp onding and matrix thus .... Row 1 e-breve sub 1 0 Row 2 0 minus breve-e sub 1 . period .... The same \noindentcannotDefinition b e expressed 4 . by 1 multiplication 2 . . transformationSince in t erm we of have complex three numbers different .... EPH should orthogonality involve a complex b etween conjugation cycles and there thus are also three cannotdifferent b e expressed by multiplication period \ hspaceSince we∗{\ havef i l lthree}1 .different\quad EPHInversion orthogonality in a b etween cycle cycles $ C there ˆ{ s are} also{\ threesigma different}$ sends a point $ p $ to the second point $ pinversions ˆ{\prime : }$ of intersection of all cycles Proposition 4 period 14 period .. A .. cycle C-tilde subinversions breve-sigma : to the power of s .. is orthogonal to a cycle C sub sigma-breve to the power\ centerline of s .. if{ fororthogonal any point u sub t o 1 e $ sub C 0 ˆ plus{ s v} sub{\ 1sigma e sub 1 in}$ tilde-C and sub passing sigma-breve through to the power $ p of . s $ } Proposition 4 . 14 . A cycle C˜s is orthogonal to a cycle Cs if for any point the cycle C-tilde sub breve-sigma to the power of s is alsoσ˘ passing through its image σ˘ u e + v e ∈ C˜s the cycle C˜s is also passing through its image \ hspaceEquation:1 ∗{\0 openf1 i l1 l } parenthesis2σ˘ . \quad 4 periodReflectionσ˘ 9 closing in parenthesis a cycle .. u $ sub C 2 ˆ e{ subs 0} plus{\sigma v sub 2 e}$ sub i1 = s Row givenby 1 le sub 0 $Mˆ plus sre{ − sub 11 m} RowRM 2 $ kwhere le sub 0 $M$ plus sre sub sends 1 . open the parenthesis cycle $C u sub ˆ{ 1 es sub} {\ 0 plussigma v sub} 1$ e sub into 1 closing the horizontal parenthesis   under the M o-dieresis bius transform defined byle the0 + sre matrix1 C subm sigma-breve to the power of period to the power of s .... Thus the point \ centerline { axis and $ Ru2 $e0 + isv2e the1 = mirror reflection in(u that1e0 + v axis1e1) . } (4.9) u sub 2 e sub 0 plus v sub 2 e sub 1 = C sub breve-sigmak to the le0 power+ sre of1 s open parenthesis u sub 1 e sub 0 plus v sub 1 e sub 1 closing parenthesis .. is the inversion of u sub 1 e sub 0s plus v sub 1 e sub 1 in C sub sigma-breve to the power of period to \ hspaceunder∗{\ f the i l l M}We o¨\quadbius transformare \quad definedgoing by\quad the matrixto \quadCσ˘. see \quad that \quadThusi n the v e r point s i o n s \quad are \quad given \quad by \quad ( 4 . 7 ) \quad and \quad reflections \quad are \quad expressed the power of s s u2e0 + v2e1 = Cσ˘ (u1e0+ Proof period .. The symbolic calculations done by G is NaC open square bracket 4 6 comma S 3 period 3 period 2 closing square bracket v1e1) is the inversion of u1e0 + v1e1 in C . Proof . The symbolic calculations done by period\noindent blacksquarethrough ( 4 . 8 ) , thus they areσ˘ essentially the same in light of Lemma 4 . 1 1 . G i NaC [ 4 6 , § 3.3.2].  \noindent Remark 4 . 1 3 . \ h f i l l Here i s a simple example where usage of complex \ h f i l l ( dual or double ) \ h f i l l numbers

\noindent i s weaker then Clifford algebras , \ h f i l l see Remark 2 . 2 . \ h f i l l A reflection of a cycle in the axis $ v = 0 $

\noindent i s represented by the conjugation ( 4 . 8 ) with the corresp onding matrix \ h f i l l $\ l e f t (\ begin { array }{ cc }\breve{e} { 1 } & 0 \\ 0 & − \breve{e} { 1 }\end{ array }\ right ) . $ \ h f i l l The same

\noindent transformation in t erm of complex numbers \ h f i l l should involve a complex conjugation and thus

\noindent cannot b e expressed by multiplication .

\ hspace ∗{\ f i l l } Since we have three different EPH orthogonality b etween cycles there are also three different

\ begin { a l i g n ∗} inversions : \end{ a l i g n ∗}

\noindent Proposition 4 . 14 . \quad A \quad c y c l e $ \ tilde {C} ˆ{ s } {\breve{\sigma}}$ \quad is orthogonal to a cycle $ C ˆ{ s } {\breve{\sigma}}$ \quad if for any point $ u { 1 } e { 0 } + v { 1 } e { 1 }\ in \ tilde {C} ˆ{ s } {\breve{\sigma}}$ the c y c l e $ \ tilde {C} ˆ{ s } {\breve{\sigma}}$ is also passing through its image

\ begin { a l i g n ∗} u { 2 } e { 0 } + v { 2 } e { 1 } = \ l e f t (\ begin { array }{ cc } l e { 0 } + s r e { 1 } & m \\ k & l e { 0 } + s r e { 1 }\end{ array }\ right ) ( u { 1 } e { 0 } + v { 1 } e { 1 } ) \ tag ∗{$ ( 4 . 9 ) $} \end{ a l i g n ∗}

\noindent under the M $ \ddot{o} $ bius transform defined by the matrix $ C ˆ{ s } {\breve{\sigma} ˆ{ . }}$ \ h f i l l Thus the point $ u { 2 } e { 0 } + v { 2 } e { 1 } = C ˆ{ s } {\breve{\sigma}} ( u { 1 } e { 0 } + $

\noindent $ v { 1 } e { 1 } ) $ \quad is the inversion of $ u { 1 } e { 0 } + v { 1 } e { 1 }$ in $ C ˆ{ s } {\breve{\sigma} ˆ{ . }}$ Proof . \quad The symbolic calculations done by G i NaC [ 4 6 , \S $ 3 . 3 . 2 ] . \ blacksquare $ Erlangen Program at Large hyphen 1 : .... Geometry of Invariants .... 2 1 \noindenthlineErlangenErlangen Program Program at Large at - Large 1 :− 1 : \ h f i l l GeometryGeometry of of Invariants Invariants \ h f i l l 2 1 2 1 Proposition 4 period 1 5 period .... The reflection 4 period 1 2 period 2 of a zero hyphen radius cycle Z sub sigma-breve to the power of s in\ [ a\ cycler u l e C{3em sub}{ breve-sigma0.4 pt }\ to] the power of s is given by the

conjugation : C sub sigma-breve to the power of s Z sub breve-sigma to the power of s C sub sigma-breves to the power of period to the Proposition 4 . 1 5 . The reflection 4 . 1 2 . 2 of a zero - radius cycle Zσ˘ in a cycle power ofs s \noindentProofCσ˘ periodis givenProposition .... Letby the C-tilde 4 sub . 1breve-sigma 5 . \ h f to i l ltheThe power reflection of s has the property 4 . 1 2 C-tilde . 2 sub of breve-sigmaa zero − toradius the power cycle of s C $ sub Z sigma-breve ˆ{ s } {\breve{\sigma}}$ toin the a power cycle of s $Cˆ to the{ powers } of{\ tilde-Cbreve sub{\sigma sigma-breve}}$ to is the given power of by s = the R period .... Then tilde-C sub sigma-breve to the power of s R tilde-C sub sigma-breve to the power of s = C sub breve-sigma tos thes powers of period to the power of s .... Mirror reflection in the conjugation : C Z C . \ beginreal line{ a l isi g given n ∗} by the .... conjugation with R comma .... thusσ˘ theσ˘ transformationσ˘ describ ed in .... 4 period 1 2 period 2 .... i s conjugation : Cˆ{ s } {\breve{\sigma}} Z ˆ{ s } {\breve{\sigma}} C ˆ{ s } {\breve{\sigma} ˆ{ . }} a conjugation with the˜ cycles C-tilde sub breve-sigma˜s sC˜ to s the power of s˜ Rs C-tilde˜s subs breve-sigma to the power of s = C sub sigma-breve to Proof . Let C has the property C C = . Then C C = C . Mirror reflection in the the\end power{ a l i ofg n s∗} and thus coincideσ˘ with open parenthesisσ˘ σ˘ 4 periodσ˘ R 9 closing parenthesisσ˘ R σ˘ σ˘ period blacksquare real line is given by the conjugation with R, thus the transformation describ ed in 4 . 1 2 . The2 cycle C-tilde sub breve-sigma to the power of s from the above proof can be characterised as follows periodi s \noindentLemma .. 4Proof period .16\ periodh f i l l ..Let C sub $ \ sigma-brevetilde {C} toˆ{ thes power} {\ ofbreve s = open{\sigma parenthesis}}$ k comma has the l comma property n comma $ m\ closingtilde parenthesis{C} ˆ{ s } {\breve{\sigma}} a conjugation with the cycle C˜s C˜s = Cs and thus coincide with (4.9). The cycle C˜s from the C.. beˆ{ ..s a} ..{\ cyclebreve .. and{\sigma for breve-sigma}}ˆ{\σ˘tildeR equal-negationslashσ˘ {C}}σ˘ ˆ{ s } 0{\ .. thebreve C-tilde{\sigma sub breve-sigma}} = to R the .σ power˘ $ \ ofh fs i .. l l beThen given ..$ \ bytilde .. open{C} ˆ{ s } {\breve{\sigma}} above proof can be characterised as follows . Lemma 4 . 16 . Let Cs = (k, l, n, m) Rparenthesis\ tilde k{ commaC} ˆ{ l commas } {\ nbreve plusminux{\sigma}} = C ˆ{ s } {\breve{\sigma} ˆσ˘{ . }}$ \ h f i l l Mirror reflection in the σ˘ 6= 0 C˜s (k, l, n± Equation:be radicalbig-linea cycle of determinantand for C sub sigma-brevethe toσ˘ thebe power given of sigma by sub comma m closing parenthesis period .. Then 1 period C-tilde\noindent sub breve-sigmareal line to the is power given of by s C the sub sigma-breve\ h f i l l conjugation to the powerof with s to the $ R power , of $ tilde-C\ h f isub l l sigma-brevethus the to transformation the power of s = R describ ed in \ h f i l l 4 . 1 2 . 2 \ h f i l l i s and tilde-C sub sigma-breve to the power of s R tilde-C sub sigma-breve to the power of s = C sub breve-sigma to the power of s semicolon T hen pdet Cσ m). \noindent2 period C-tildea conjugation sub breve-sigma with to the the power cycle of s .. $ and\ tilde C sub{C sigma-breve} ˆ{ s } to{\ thebreve power{\ ofsigma s have}} commonR σ˘ , roots\ tilde semicolon{C} ˆ{ s } {\breve{\sigma}} = C ˆ{ s } {\breve{\sigma}}$ andthuscoincidewith˜s sC˜ s ˜s ˜s s $( 4 . 9 ) . \ blacksquare $ 3 period .. in the sigma-breve hyphen1 implementation. Cσ˘ Cσ˘ σ˘ = R theand cycleCσ˘ RC Cσ˘ = subCσ˘ breve-sigma; to the power of s passes the centre of C-tilde sub breve-sigmaThe c y c l eto the $ \ powertilde of{ periodC} ˆ{ tos the} power{\breve of s {\sigma}}$ from the above proof can be characterised as follows . Lemma \quad 4 . 16 . \quad Let˜s $ C ˆ{ s s} {\breve{\sigma}} =(k,l,n,m)$ Proof period .... This i s calculated2. byC Gσ˘ i NaCand ....C openσ˘ have square common bracket roots 4 6 comma ; .... S 3 period 3 period 5 closing square bracket period ....\quad Alsobe one\ canquad directa \ observequad 4c y period c l e \ 1quad 6 periodand 2 for f o r real $ \breve{\ssigma}\ne 0 $ \˜quads the $ \ tilde {C} ˆ{ s } {\breve{\sigma}}$ 3 . in the σ˘− implementation the cycle Cσ˘ passes the centre of Cσ˘. \quadrootsProofbe comma given . sinceThis\quad they i sareby calculated fixed\quad points by$ of (G the i NaCinversion k ,[ 4 period 6 l , § ....3 , . Also 3 n . 5the ]\pm .transformation Also$ one can of C direct sub sigma-breve observe 4 to the power of s t o a flat cycle . 1 6 . 2 for real \ beginimplies{ a that l i g n C∗} sub sigma-breve to the power of s i s passing the centre of inversion comma hence 4 periods 16 period 3 period blacksquare roots , since they are fixed points of the inversion . Also the transformation of Cσ˘ t o a flat \ tagIn∗{ open$\ sqrt square{\ bracketdet 7C 2 comma ˆ{\sigma S 10 closing} {\ squarebreve bracket{\sigma the}}} inversion{ , of} secondm kind ) related . $} tThen o a parabola\\ 1 v =. k open\ tilde parenthesis{C} ˆ{ us } {\breve{\sigma}} C ˆ{ cycles } {\breve{\sigma}}ˆ{\ tilde {C}}ˆ{ s } {\breve{\sigma}} = R and \ tilde {C} ˆ{ s } {\breve{\sigma}} minus limplies closing parenthesis that Cs i to s passing the power the of 2 centre plus m of was inversion defined , hence 4.16.3. R \ tilde {C} ˆ{ sσ˘ } {\breve{\sigma}} = C ˆ{ s } {\breve{\sigma}} ; by the mapIn : [ 7 2 , §10 ] the inversion of second kind related t o a parabola v = k(u − l)2 + m was \endEquation:{defineda l i g n ∗} open parenthesis 4 period 10 closing parenthesis .. open parenthesis u comma v closing parenthesis arrowright-mapsto open parenthesisby the u comma map : 2 open parenthesis k open parenthesis u minus l closing parenthesis to the power of 2 plus m closing parenthesis minus v closing\ centerline parenthesis{ $ comma 2 . \ tilde {C} ˆ{ s } {\breve{\sigma}}$ \quad and $ C ˆ{ s } {\breve{\sigma}}$ have common roots ; } i period e period .. the .. parabola bisects .. the vertical line j oining .. a point .. and its .. image period .. Here i s .. the result \ centerlineexpression this{3 transformation . \quad in throughthe $ the\(breveu, usual v) 7→{\ inversion(u,sigma2(k(u} in− l parabolas)2 −+$m) − implementation :v), the cycle $(4 C.10) ˆ{ s } {\breve{\sigma}}$ passesProposition the centre 4 period 1 of 7 period $ \ tilde .... The{C inversion} ˆ{ s of} s{\ econdbreve kind{\ opensigma parenthesis} ˆ{ . 4}} period$ } 1 0 closing parenthesis is a composition of three inversionsi . : e in . the parabola bisects the vertical line j oining a point and its image . \noindentparabolasHere i uProof s to the the . power result\ h f i of l l 2expressionThis minus 2i lu s this minus calculated transformation 4 mv minus by G m slash i through NaC k =\ h 0the f comma i l lusual[ u4 inversion to 6 the , \ powerh f iin l l of parabolas\ 2S minus3 . 32 lu . minus 5 ] . m\ slashh f i l k l =Also 0 one can direct observe 4 . 1 6 . 2 for real comma: .. and the real line period \noindentProofProposition periodroots .... See , 4 symbolic since . 1 7 they .calculationThe are inversion in fixed open square points of s bracket econd of the 4kind 6 comma inversion( 4 .S 1 3 period0 ) . is\ 3ah periodf composition i l l Also 6 closing the of square three transformation bracket period blacksquare of $ C ˆ{ s } {\breve{\sigma}}$ t oRemark ainversions flat 4 period cycle : 18 in period .... Yaglom in open square bracket 7 2 comma S 1 0 closing square bracket considers the usual inversion open parenthesisparabolas quotedblleftu2 − of2lu the− first4mv kind− m/k quotedblright= 0, u2 − 2lu closing− m/k parenthesis= 0, and only the in real line . \noindent implies that $ C ˆ{ s } {\breve{\sigma}}$ i s passing the centre of inversion , hence $ 4 degeneratedProof . parabolas open parenthesis quotedblleft parabolicSee circles symbolic quotedblright calculation closing in parenthesis [ 4 6 , § of3.3 the.6]. form u to the power of 2 minus 2. lu plus 16Remark m = . 0 period 3 4 . 18 ... However . \ blacksquareYaglom the inversion in [ 7 $ 2 , §1 0 ] considers the usual inversion ( “ of the first kind of the” ) second only in kind requires for it s decomposition like in Proposition 4 period 1 7 .. at least one inversion \ hspacein adegenerated proper∗{\ f parabolic i l l } In parabolas [ cycle 7 2 u to , ( the\ “S parabolic power10 ] of the 2 minuscircles inversion 2” lu ) minus of of the 2 second nv form plus mu kind2 =− 02 periodlu related+ m ..= Thus 0 t. o suchHowever a parabola inversions the are $ indeed v = of another k ( u kind−inversion withinl Yaglom ) of ˆ{ the2 quoteright} second+ s m$ kind framework requires was open defined for square it s bracket decomposition 7 2 closing like square in bracketProposition comma 4 but . 1 are 7 not at in our period Anotherleast important one inversion difference b etween inversions from open square bracket 7 2 closing square bracket and our wider set of transforma hyphen \noindentt ionsin open a properby parenthesis the parabolic map 4 period : cycle 7 closingu2 − parenthesis2lu − 2nv + ism what= 0. quotedblleftThus such special inversions quotedblright are indeed .. open parenthesisof another vertical closing parenthesis lines doeskind notwithin form an Yaglom invariant ’ set s framework comma as can [ 7 b 2 e seen] , but from are not in our . \ beginFig period{ a l i g 9 n open∗} Another parenthesis important c closing parenthesis difference comma b etween and inversions thus they are from not [.. 7 quotedblleft 2 ] and our special wider quotedblright set of lines anymore period (4 periodtransforma u 3 , .. Focal v - orthogonality ) \mapsto ( u , 2 ( k ( u − l ) ˆ{ 2 } + m ) − v ) , \Ittag ist natural∗{ ions$ ( ( 4 to . 4consider 7 ) is . what invariants 10 “ special ) of $} higher ” orders ( vertical which ) are lines generated does not by openform parenthesis an invariant 4 period set , 1 as closing can b parenthesis period .. Such invariants\end{ea l seen i g n ∗} from Fig . 9 ( c ) , and thus they are not “ special ” lines anymore . shall4 have . 3 at least Focal one orthogonalityof the following properties \noindentbulletIt is.. contains naturali . e a to . non\ considerquad hyphenthe linearinvariants\quad powerparabola of of higherthe same ordersbisects cycle semicolon which\quad arethe generated vertical by ( line 4 . 1 j ) .oining Such\quad a point \quad and i t s \quad image . \quad Here i s \quad the r e s u l t expressionbulletinvariants .. accommodate this shall transformation have more thanat least two cyclesone through of period the following the usual properties inversion in parabolas : The consideration of higher• ordercontains invariants a non is similar - linear t o a power transition of the from same Riemannian cycle ; geometry \noindentt o FinslerProposition one open square 4 bracket . 1• 7 1accommodate ... 4\ h comma f i l l The .. 2 2 more inversion comma than .. 6 two 1 of closing cycles s econd square . kind bracket ( period 4 . 1 0 ) is a composition of three inversions : in It iThe s interesting consideration that higher of higher order invariants order invariants is similar t o a transition from Riemannian geom- \noindentetry tparabolas o Finsler one $ [ u 1 ˆ{ 42 ,} −2 2 ,2 6 1lu ] . − 4 mv − m / k = 0 , u ˆ{ 2 } − 2 lu − m / k = 0It i s , interesting $ \quad thatand higherthe real order line invariants .

\noindent Proof . \ h f i l l See symbolic calculation in [ 4 6 , \S $ 3 . 3 . 6 ] . \ blacksquare $

\noindent Remark 4 . 18 . \ h f i l l Yaglom in [ 7 2 , \S 1 0 ] considers the usual inversion ( ‘‘ of the first kind ’’ ) only in

\noindent degenerated parabolas ( ‘‘ parabolic circles ’’ ) of the form $ u ˆ{ 2 } − 2 lu + m = 0 . $ \quad However the inversion of the second kind requires for it s decomposition like in Proposition 4 . 1 7 \quad at least one inversion

\noindent in a proper parabolic cycle $ u ˆ{ 2 } − 2 lu − 2 nv + m = 0 . $ \quad Thus such inversions are indeed of another kind within Yaglom ’ s framework [ 7 2 ] , but are not in our .

\ hspace ∗{\ f i l l }Another important difference b etween inversions from [ 7 2 ] and our wider set of transforma −

\noindent t ions ( 4 . 7 ) is what ‘‘ special ’’ \quad ( vertical ) lines does not form an invariant set , as can b e seen from Fig . 9 ( c ) , and thus they are not \quad ‘‘ special ’’ lines anymore .

\noindent 4 . 3 \quad Focal orthogonality

\noindent It is natural to consider invariants of higher orders which are generated by ( 4 . 1 ) . \quad Such invariants shall have at least one of the following properties

\ centerline { $ \ bullet $ \quad contains a non − linear power of the same cycle ; }

\ centerline { $ \ bullet $ \quad accommodate more than two cycles . }

\noindent The consideration of higher order invariants is similar t o a transition from Riemannian geometry t o Finsler one [ 1 \quad 4 , \quad 2 2 , \quad 6 1 ] .

\ centerline { It i s interesting that higher order invariants } 22 .... V period V period Kisil \noindenthline22 22 \ h f i l l V . V . K i s i l V . V . Kisil Figure 9 period .... Three types of inversions of the rectangular grid period The initial rectangular grid open parenthesis a closing parenthesis is\ [ inverted\ r u l e {3em}{0.4 pt }\ ] elliptically in the unit circle open parenthesis shown in red closing parenthesis on open parenthesis b closing parenthesis comma parabolically on openFigure parenthesis 9 . cThree closing types parenthesis of inversions and hyperbolically of the rectangular on open grid parenthesis . The initial d closing rectangular parenthesis grid ( a period ) is inverted .... The \noindentblueelliptically cycleFigure open in parenthesis the 9 unit . \ circleh collapsed f i l ( l shownThree to in a red typespoint ) on at (of the b ) inversions origin, parabolically on open of on parenthesis the( c ) and rectangular hyperbolicallyb closing parenthesis grid on ( .d ) Theclosing . The initial parenthesis rectangular represent the grid ( a ) is inverted image ofblue the cycle cycle ( at collapsed infinity under to a point at the origin on ( b ) ) represent the image of the cycle at infinity under \noindentinversioninversion periodelliptically . in the unit circle ( shown in red ) on ( b ) , parabolically on ( c ) and hyperbolically on ( d ) . \ h f i l l The 1 period .. can b e built on1 . top of can the b already e built defined on top ones of semicolon the already defined ones ; \noindent2 period ..blue can produce cycle lower ( collapsed order2 . invariants can to produce a period point lower at the order origin invariants on ( . b ) ) represent the image of the cycle at infinity under i nFor v eFor r each s i o eachn ofthe . of two the above two transitions above transitions we consider an we example consider period an We example already . know We that already a similarity know that a of asimilarity cycle with another cycle i s a new cycle open parenthesis 4 period 8 closing parenthesis period .... The inner product of lat er with a third\ centerline givenof a cycle{1 with . \quad anothercan cycle b e ibuilt s a new on cycle top of( 4 the . 8 ) already . The defined inner product ones ; of} lat er with a cyclethird form given a j oint invariant of those three cycles : \ centerlineEquation:cycle form open{2 .parenthesis a\ jquad oint invariantcan 4 period produce of1 1 those closing lower three parenthesis order cycles invariants .. : angbracketleft . C} sub sigma-breve 1 to the power of s C sub breve-sigma 2 to the power of s C sub sigma-breve 1 to the power of comma to the power of s C sub breve-sigma 3 to the power of s right angbracket comma \noindent For each of the two above transitions we consider an example . We already know that a similarity which is build from the second hyphen order invariants s angbracketlefts s times comma times right angbracket period Now we can reduce the hC C C , C i, (4.11) order of this invariant σ˘1 σ˘2 σ˘1 σ˘3 \noindent of a cycle with another cycle i s a new cycle ( 4 . 8 ) . \ h f i l l The inner product of lat er with a third given by fixingwhich C is sub build sigma-breve from the 3 to second the power - orderof s b e invariant the real lineh· open, ·i. Now parenthesis we can which reduce is it self the invariant order of closing this parenthesis period .. The obtained invariant of two cycles \noindentinvariantcycle form a j oint invariant of those three cycles : deserves a sp ecials consideration period .. Alternatively it emerges from Definitions 4 period 1 and 3 period 1 5 period by fixing Cσ˘3 b e the real line ( which is it self invariant ) . The obtained invariant of two Definitioncycles 4 deservesperiod 19 period a sp ecialThe focal consideration orthogonality . open Alternatively parenthesis f hyphen it emerges orthogonality from closing Definitions parenthesis 4 . of 1 a cycle C sub sigma-breve to\ begin the power{ a l i gof n s∗} to a cycle tilde-C sub sigma-breve to the power of s i s defined \ langleand 3 .C 1 ˆ 5{ . s } {\breve{\sigma} 1 } C ˆ{ s } {\breve{\sigma} 2 } C ˆ{ s } {\breve{\sigma} by the condition that the cycle C sub sigma-breve to the power of s to the power of tilde-Cs sub sigma-breve˜s to the power of s C sub 1 ˆ{ Definition, }} C ˆ{ 4s .} 19{\ . breveThe focal{\sigma orthogonality} 3 }\(rangle f - orthogonality, \ tag ∗{ ) of$ a ( cycle 4Cσ˘ .to 1a cycle 1C )σ˘ $} breve-sigma to the power of s i s orthogonal open parenthesissC˜ in s thes sense of Definition 4 period 1 closing parenthesis t o the real \end{ia sl i definedg n ∗} by the condition that the cycle Cσ˘ σ˘ Cσ˘ i s orthogonal ( in the sense of Definition line4 comma . 1 ) i t period o the e real i s a self line hyphen , i . adje i oints a cycleself - in adj the oint sense cycle of Definition in the 3 sense period of 1 5Definition period .. Analytically 3 . 1 5 . this is defined by Equation:Analytically open parenthesis this is defined 4 period by 1 2 closing parenthesis .. Re tr open parenthesis C sub sigma-breve to the power of s to the power of tilde-C\noindent sub sigma-brevewhich is to build the power from of s the C sub second breve-sigma− order to the invariant power of s R sub $ \ sigma-brevelangle to\cdot the power, of s\ closingcdot parenthesis\rangle = 0 . $ Nowand we we can denote reduce it by C the sub order sigma-breve of this to the invariant power of s turnstileright tilde-C sub sigma-breve to the power of period to the power of s Remark 4 period 20 period .... This definition i s explicitlysC˜ s s baseds on the invariance of the real line and i s an

\ begin { a l i g n ∗} \Re t r ( C ˆ{ s } {\breve{\sigma}}ˆ{\ tilde {C}}ˆ{ s } {\breve{\sigma}} C ˆ{ s } {\breve{\sigma}} R ˆ{ s } {\breve{\sigma}} ) = 0 \ tag ∗{$ ( 4 . 1 2 ) $} \end{ a l i g n ∗}

\noindent and we denote it by $ C ˆ{ s } {\breve{\sigma}}\dashv \ tilde {C} ˆ{ s } {\breve{\sigma} ˆ{ . }}$

\noindent Remark 4 . 20 . \ h f i l l This definition i s explicitly based on the invariance of the real line and i s an

\noindent illustration to the boundary value effect from Remark 3 . 5 . Remark 4 . 2 1 . \quad It i s easy t o observe the following

\ centerline {1 . \quad f − orthogonality i s not a symmetric $ : C ˆ{ s } {\breve{\sigma}}\dashv \ tilde {C} ˆ{ s } {\breve{\sigma}}$ does not implies $ \ tilde {C} ˆ{ s } {\breve{\sigma}}\dashv C ˆ{ s } {\breve{\sigma}} ; $ } Erlangen Program at Large hyphen 1 : .... Geometry of Invariants .... 23 \noindenthlineErlangenErlangen Program Program at Large at - Large 1 :− 1 : \ h f i l l GeometryGeometry of of Invariants Invariants \ h f i l l 23 23 2 period .. since the real axis R and orthogonality open parenthesis 4 period 2 closing parenthesis are SL sub 2 open parenthesis R closing parenthesis\ [ \ r u l e { hyphen3em}{0.4 invariant pt }\ obj] ects f hyphen orthogonality is also SL sub 2 open parenthesis R closing parenthesis hyphen invariant period However an2 . invariance since of the f hyphen real axis orthogonalityR and orthogonality under inversion ( of4 .cycles 2 ) are requiredSL2( someR)− studyinvariant since obj ects f - \ hspacetheorthogonality real∗{\ linef i is l l not}2 an . invariant\quad since of such the transformations real axis in general $R$ period and orthogonality ( 4 . 2 ) are $ SL { 2 } (R ) Lemma− $ 4 invariant period 22 period obj .. ects The image f − orthogonality Cis sub also sigma-breveSL2(R)− toinvariant the power . of s sub 1 R sub breve-sigma to the power of s C sub sigma-breve to the powerHowever of s sub 1 ..an of invariance the real line of under f - orthogonality inversion in C sub under breve-sigma inversion to the of cycles power of required s sub 1 = some open study parenthesis k comma l comma n comma\ centerlinesince m closing{ i sparenthesis a l s o $ is SL the { 2 } (R) − $ invariant . } cyclethe : open real parenthesisline is not 2 an ss invariant sub 1 to the of power such transformationsof sigma-breve kn comma in general 2 ss sub . Lemma 1 to the power 4 . 22 of breve-sigma . The ln comma s to the power Cs1 Rs Cs1 Cs1 = (k, l, n, m) of\ hspace 2 openimage∗{\ parenthesisf i l lσ˘}Howeverσ l˘ toσ˘ theof power an the invariance of real 2 plus line sigma-breve under of f inversion− northogonality to the in powerσ˘ of 2 minus under mk inversion closingis the parenthesis of cycles comma required 2 ss sub 1 tosome the power study of since breve-sigma mn closing parenthesis period \noindentIt is the realthe line real if s times line det is open not parenthesis an invariant C sub sigma-breve of such transformations to the power of s sub in 1 closing general parenthesis . negationslash-equal 0 and cycle : eitherLemma 4 . 22 . \quad The image $ C ˆ{ s { 1 }} {\breve{\sigma}} R ˆ{ s } {\breve{\sigma}} C ˆ{ s { 1 }} {\breve{\sigma}}$ \quad of the real line underσ˘ inversionσ˘ in2 $2 C ˆ{ 2 s { 1 }} σ˘{\breve{\sigma}} = ( k , l , n 1 period s sub 1 n = 0 comma ..(2 inss1 thiskn, ..2 casess1 ln .., it.. s ( isl ..+ aσn ˘ .. composition− mk), 2ss ..1 mn of SL). sub 2 open parenthesis R closing parenthesis hyphen action, m .. by ) .. $ Row i s 1 l the minus me sub 0 Row 2 ke sub 0 minus l . .. and .. the s1 re f-lIt ection is the in real the linereal line if semicolons· det (C orσ˘ ) 6= 0 and either \ begin { a l1 i g. n s∗}1n = 0, in this case it is a composition of SL2(R)− action 2 period sigma-breve = 0 comma i period e period the parabolic case of the cycle space period c yIf c this lby e .... : condition\\ l(− isme 2 satisfied0 s s thanand{ 1 f hyphen}ˆ{\ the breve orthogonality{\sigma preserved}} kn .... by , the ....2 inversion s s { C-tilde1 }ˆ{\ subbreve breve-sigma{\sigma to the}}\ powerln of s right, arrows ˆ{ C2 sub} sigma-breve(ke l0 ˆ{− to2l the} power+ \ ofbreve s sub{\ 1 tosigma the power} n of ˆ tilde-C{ 2 } sub − sigma-brevemk ) to the , power 2 of s s s C sub{ 1 breve-sigma}ˆ{\breve to the{\sigma power}} of s mnsub 1 ) . re f − l ection in the real line ; or \endin{ C-tildea l i g n sub∗} breve-sigma2. toσ the˘ = power 0, i . ofe . period the parabolic to the power case of sof the cycle space . TheIf following this explicit condition expressions is satisfied of f hyphen than orthogonality f - orthogonality reveal further preserved connections with by the cycles quoteright inversion ˜ \noindent˜s It iss1 C the s s1 real line if $ s \cdot $ det $ ( C ˆ{ s { 1 }} {\breve{\sigma}} ) \not= invariantsCσ˘ → periodCσ˘ σ˘ Cσ˘ 0 $Proposition and either 4 period 23 period .... f hyphen orthogonality .... of C sub p to the power of s .... to C-tilde sub p to the power of s .... is given by either of the following equivalent \ hspace ∗{\ f i l l } $ 1 . s { 1 } n = 0˜s , $ \quad in t h i s \quad case \quad i t \quad i s \quad a \quad composition \quad o f identities inCσ˘. $ SLLine{ 1 n-tilde2 } open(R) parenthesis− l to$ the a power c t i o n of\ 2quad minusby breve-e\quad sub 1$ to\ thel e f powert (\ begin of 2 n{ toarray the power}{ cc } ofl 2 minus & − mk closingme { parenthesis0 }\\ ke plus { 0 } & − l \endThe{ array following}\ right explicit)$ expressions\quad and of\quad f - orthogonalitythe reveal further connections with cycles m-tilde’ nk invariants minus 2 l-tilde . nl plus k-tilde sub mn = 0 comma or Line 2 n-tilde determinant open parenthesis C sub breve-sigma to the power of s closing parenthesis plus n angbracketleftBig C sub p to the power of comma to the power of s C-tilde sub p to the power of s angbracketrightBig Proposition 4 . 23 . f - orthogonality of Cs to C˜s is given by either of the following =\ centerline 0 period { re $ f−l $ ection in the real linep ; or }p Proofequivalent period .... This i s another G i NaC calculation open square bracket 4 6 comma S 3 period 4 period 1 closing square bracket period blacksquare\ centerlineidentities{ $ 2 . \breve{\sigma} = 0 , $ i . e . the parabolic case of the cycle space . } The f hyphen orthogonality may b e again related t o the usual orthogonality through an appropriately n˜(l2 − e˘2n2 − mk) +mnk ˜ − 2˜lnl + k˜ = 0, or \noindentchosen f hyphenI f t h ghost i s \ cycleh f i l lcommacondition compare1 is the satisfied next proposition than with fmn− Propositionorthogonality .. 4 period preserved 9 : \ h f i l l by the \ h f i l l i n v e r s i o n $ \ tilde {C} ˆ{ s } {\breve{\sigma}}\rightarrows C ˆs{ ˜s { 1 }} {\breve{\sigma}}ˆ{\ tilde {C}}ˆ{ s } {\breve{\sigma}} Proposition 4 period 24 period .... Let C sub sigma-breven˜ det( toC theσ˘ ) + powernhCp, ofCp si be= 0 a. cycle comma then its f hyphen ghost cycle tilde-C sub Csigma-breve ˆ{ s { to1 the}} power{\breve of breve-sigma{\sigma =}} C$ sub sigma-breve to the power of chi open parenthesis sigma closing parenthesis R sub sigma-breve Proof . This i s another G i NaC calculation [ 4 6 , § 3.4.1]. to the power of breve-sigma C sub breve-sigma to the power of chi open parenthesis sigma closing parenthesis is the reflection \ beginThe{ a l f i g- northogonality∗} may b e again related t o the usual orthogonality through an appropriately of thechosen real linef - in ghost C sub cycle sigma-breve, compare to the the power next of chi proposition open parenthesis with sigma Proposition closing parenthesis 4 . 9 : comma where chi open parenthesis sigma closingin parenthesis\ tilde {C} .. isˆ{ thes Heaviside} {\breve function{\sigma 4 period} ˆ 6{ period. }} .. Then \end{ a l i g n ∗} s ˜σ˘ χ(σ) σ˘ χ(σ) 1 periodProposition .. Cycles C 4 sub . 24 sigma-breve . Let toCσ˘ thebe power a cycle of ,s then.. and its tilde-Cf - ghost sub sigma-breve cycle Cσ˘ to= C theσ˘ powerRσ˘ Cσ˘ of breve-sigmais the have the same roots period reflection The following explicitχ expressions(σ) of f − orthogonality reveal further connections with cycles ’ 2 periodof the chi real open line parenthesis in Cσ˘ sigma, where closingχ parenthesis(σ) is the hyphenHeaviside Centre of function C-tilde sub 4 . breve-sigma 6 . Then to the power of sigma-breve coincides with invariants . s ˜σ˘ the breve-sigma hyphen focus1 of . C sub Cycles sigma-breveCσ˘ toand the powerCσ˘ ofhave comma the tosame the power roots of. s .. consequently all lines f hyphen orthogonal to ˜σ˘ s C sub sigma-breve2. χ(σ)− toCentre the power of ofC sσ˘ ..coincides are passing with one of the its fociσ˘− periodfocus of Cσ˘, consequently all lines f - \noindent3 periodorthogonal s hyphenProposition to Reflection 4 inversion . 23 . sub\ h fin i l sigma-breve l f − orthogonality from breve-sigma\ h period f i l l too f in C-tilde $ C ˆ{ C subs } breve-sigma{ p }$ \ toh the f i l powerl to of $ s defined\ tilde {C} ˆ{ s } { p }$ \ h f i l l is given by either of thes following equivalent from f hyphen orthogonality open parenthesisCσ˘ sare ee Definition passing one4 period of its 1 2 foci period . 1 closing parenthesis .. coincides with usual Proof period3. s ....− Reflectioninversion This again i s calculatedσ˘σ.˘ in G i NaC comma see open square bracket 4 6 comma S 3 period 4 period 3 closing square bracket in inCC˜ s defined from f - orthogonality ( s ee Definition 4 . 1 2 . \noindent i d e n t i t i e s σ˘ period1 blacksquare ) coincides with usual For the reason .... 4 period 2 4 period 2 this relation b etween cycles may b e labelled as focal orthogonality comma .... cf period Proof . This again i s calculated in G i NaC , see [ 4 6 , § 3.4.3]. \ [ \withbegin ....{ 4a l period i g n e d 9}\ periodtilde 1 period{n} ....( It can l ˆ generates{ 2 } − the corresp \breve onding{e} ˆ inversion{ 2 } { similar1 } tn o Definition ˆ{ 2 } 4 − period mk 1 2 period ) + 1 which\ tilde ob {m} nk −For the2 reason\ tilde 4{ .l } 2 4 .nl 2 this + relation\ tilde b{ etweenk} { cyclesmn } may= b 0 e labelled , or as\\focal orthogonality hyphen, cf . \viouslytilde { reducesn}\ todet the usual( inversion C ˆ{ s in} the{\ f hyphenbreve ghost{\sigma cycle}} period) .... + The extravagant n \ langle f hyphenC orthogonality ˆ{ s } { p will ˆ{ , }}\ tilde {C} ˆ{ s } { p } \ranglewith= 4 . 9 0 . 1 . . \ Itend can{ a generatesl i g n e d }\ ] the corresp onding inversion similar t o Definition 4 . 1 2 . unexpectedly1 which ob appear - again from consideration of length and distances in the next section and i s usefulviously for reduces infinitesimal to the cycles usual Section inversion 6 period in 1 theperiod f - ghost cycle . The extravagant f - \noindentorthogonalityProof . will\ h f i l l This i s another G i NaC calculation [ 4 6 , \S $ 3 . 4 . 1 ] . \ blacksquare $ unexpectedly appear again from consideration of length and distances in the next section and \noindenti s usefulThe for f − infinitesimalorthogonality cycles may Section b e 6 again . 1 . related t o the usual orthogonality through an appropriately

\noindent chosen f − ghost cycle , compare the next proposition with Proposition \quad 4 . 9 :

\noindent Proposition 4 . 24 . \ h f i l l Let $ C ˆ{ s } {\breve{\sigma}}$ be a cycle , then its f − ghost c y c l e $ \ tilde {C} ˆ{\breve{\sigma}} {\breve{\sigma}} = C ˆ{\ chi ( \sigma ) } {\breve{\sigma}} R ˆ{\breve{\sigma}} {\breve{\sigma}} C ˆ{\ chi ( \sigma ) } {\breve{\sigma}}$ is the reflection

\noindent of the real line in $Cˆ{\ chi ( \sigma ) } {\breve{\sigma}} , $ where $ \ chi ( \sigma ) $ \quad is the Heaviside function 4 . 6 . \quad Then

\ centerline {1 . \quad Cycles $ C ˆ{ s } {\breve{\sigma}}$ \quad and $ \ tilde {C} ˆ{\breve{\sigma}} {\breve{\sigma}}$ have the same roots . }

\ hspace ∗{\ f i l l } $ 2 . \ chi ( \sigma ) − $ Centre o f $ \ tilde {C} ˆ{\breve{\sigma}} {\breve{\sigma}}$ coincides with the $ \breve{\sigma} − $ focus of $Cˆ{ s } {\breve{\sigma} ˆ{ , }}$ \quad consequently all lines f − orthogonal to

\ centerline { $ C ˆ{ s } {\breve{\sigma}}$ \quad are passing one of its foci . }

\ hspace ∗{\ f i l l } $ 3 . s − Re f l e c t i o n { i n v e r s i o n } { in }\breve{\sigma} ˆ{\breve{\sigma} . } { in {\ tilde {C}} C ˆ{ s } {\breve{\sigma}}}$ defined from f − orthogonality ( s ee Definition 4 . 1 2 . 1 ) \quad coincides with usual

\noindent Proof . \ h f i l l This again i s calculated in G i NaC , see [ 4 6 , \S $ 3 . 4 . 3 ] . \ blacksquare $

\noindent For the reason \ h f i l l 4 . 2 4 . 2 this relation b etween cycles may b e labelled as focal orthogonality , \ h f i l l c f .

\noindent with \ h f i l l 4 . 9 . 1 . \ h f i l l It can generates the corresp onding inversion similar t o Definition 4 . 1 2 . 1 which ob −

\noindent viously reduces to the usual inversion in the f − ghost cycle . \ h f i l l The extravagant f − orthogonality will

\noindent unexpectedly appear again from consideration of length and distances in the next section and

\noindent i s useful for infinitesimal cycles Section 6 . 1 . 24 .... V period V period Kisil \noindenthline24 24 \ h f i l l V . V . K i s i l V . V . Kisil Figure 10 period .. open parenthesis a closing parenthesis The square of the parabolic diameter is the square of the distance between roots if\ [ they\ r u l e {3em}{0.4 pt }\ ] are real open parenthesis z sub 1 and z sub 2 closing parenthesis comma otherwise the negative square of the distance between the adjoint roots openFigure parenthesis 10 . z( sub a ) The 3 and square z sub of 4 closingthe parabolic parenthesis diameter period is the square of the distance between roots if they \noindentopenare parenthesis realFigure (z1 and b closingz 102), otherwise . parenthesis\quad the( a negativeDistance ) The square as square extremum of the of distanceof the diameters parabolic between in elliptic the diameter adjoint open parenthesis roots is (z3 theand z sub squarez4) 1. ( and b ) z of sub the 2 closing distance parenthesis between roots if they andare parabolicDistance r e a l open $ as ( extremum parenthesis z { of1 z diameters} sub$ 3 and and in z $elliptic sub z 4 closing{ (z21 and} parenthesisz)2) and , parabolic $ cases otherwise period (z3 and z the4) cases negative . square of the distance between the adjoint roots $ (5 ..5 Metric z { Metric ..3 properties}$ and .. propertiesfrom$ z ..{ cycle4 } .. invariants) from . $ cycle invariants (So b farSo ) Distance we far discussed we discussed as only extremum invariants only invariants oflike diameters orthogonality like orthogonality in comma elliptic which , are which $ related ( are z t related o{ angles1 }$ periodt o and angles .. Now $ . z we{ turnNow2 } ) $ and parabolic $ (t owe metric z turn{ properties3 t}$ o metric and similar properties$ t z o distance{ 4 } similar period) $ t o c a distance s e s . . 5 period5 . 1 1 .. Distances Distances and lengths and lengths \noindentTheThe covariance covariance5 \quad of cyclesMetric of cyclesopen parenthesis\quad ( see Lemmap r seeo p e Lemma r t2 i e. s 1 24\quad period ) suggestsfrom 1 4 closing them\quad parenthesis asc “ y ccircles l e \ suggestsquad ” in eachithem n v a rofas i a the nquotedblleft t s EPH circles quotedblright in each ofcases the EPH . cases period \noindentThusThus we play weSo theplay far standard wethe discussed standard mathematical mathematical only game invariants : .. turn game some like: properties turn orthogonality some of classical properties objects , which of into classical are related objects t o angles . \quad Now we turn tdefinitions ointo metric definitions of properties new ones of period new similar ones . t o distance . s DefinitionDefinition 5 period 5 1 . period 1 . .... The sigma-breveThe σ˘− hyphenradius radiusof a of cycle a cycleC Cσ˘ if sub squared breve-sigma i s equal to the t power o the ofσ˘ s− if squared i s equal t o the sigma-breve\noindentdeterminant hyphen5 . 1 determinant of\quad cycleDistances ’ sof cycle quoteright and lengths s k hyphenk− normalised normalised ( open see parenthesisDefinition see 3 . Definition 9 ) matrix 3 period , i . e 9 . closing parenthesis matrix comma i period e period \noindentEquation:The open covarianceparenthesis 5 period of cycles 1 closing ( parenthesis see Lemma .. r 2 to . the 1 power 4 ) suggests of 2 = determinant them as C sub ‘‘ sigma-breve circles ’’ to the in power each of of s divided the EPH cases . by k to the power of 2 = l to the power of 2 minus breve-sigmadet Cs l2 − n toσn˘ the2 − powerkm of 2 minus km divided by k to the power of 2 sub period \noindentAs usual commaThus wethe sigma-breve play the hyphenstandard diameterr2 = mathematical ofσ˘ a= cycles i sgame two t : imes\quad it s radiusturn period some properties(5. of1) classical objects into k2 k2 . definitionsLemma 5 period of 2 new period ones .. The . sigma-breve hyphen radius of a cycle C sub breve-sigma to the power of s is equal to 1 slash k comma where kAs is usualopen parenthesis , the σ˘− 2diameter comma 1 closingof a cycles parenthesis i s two hyphen t imes entry it of s radiusdet hyphen . Lemma normalised 5 . 2 . The \noindentmatrix openDefinition parenthesis see 5 .Definition 1s . \ h 3 f periodi l l The 9 closing $ \breve parenthesis{\sigma .. of} the cycle − $ period radius of a cycle $Cˆ{ s } {\breve{\sigma}}$ σ˘− radius of a cycle Cσ˘ is equal to 1/k, where k is ( 2 , 1 ) - entry of det - normalised ifGeometrically squaredmatrix i( sinsee variousequal Definition EPH t o cases the3 . this 9 $ )\ correspbreveof the{\ ondssigma cycle t o the} . following − $ determinant of cycle ’ s openGeometrically parenthesis e comma in various h closing EPH parenthesis cases this .. The corresp value of onds open t parenthesis o the following 5 period ( 1 e closing , h ) parenthesis The value i s the usual radius of a circle or\noindent hyperbolaof ( 5 semicolon .$ 1 k ) i s− the$ usual normalised radius of ( a see circle Definition or hyperbola 3 ;. 9 ) matrix , i . e . open parenthesis( p ) p closing The parenthesisdiameter of .. The a parabola diameter i of s a the parabola ( Euclidean i s the open ) distance parenthesis b etween Euclidean it closing s ( real parenthesis ) distance b etween it\ begin s openroots{ parenthesisa l i g , n i∗} . e . real closing solu - parenthesis roots comma i period e period .. solu hyphen r ˆ{ 2 } = \ f r a c {\det C ˆ{ s } {\breve{\sigma}}}{ k ˆ{ 2 }} = \ f r a c { l ˆ2 { 2 } − \breve{\sigma} tions of ku to theku power2 − 2 oflu 2+ minusm = 0 2, lu plus m = 0 comma or roots of it s .. quotedblleft−ku adj2 + oint 2lu quotedblright+ m − 2l = ..0 parabola minus ku to the n ˆ{ 2 }tions − ofkm }{ k ˆ{ 2 }} or{ roots. }\ tag of∗{ it$ s ( “ adj 5 oint . ” 1 parabola ) $} k power of( see 2 plus Fig 2 .lu 10 plus ( am ) minus ) . Remark 2 l to the power 5 . 3 of . 2 dividedNote by that k = 0 \endopen{ a l parenthesis i g n ∗} see Fig period 10 open parenthesis a closing parenthesis closing parenthesis period Remark 5 period 3 period .. Note that 2 σ˘ = −4 ∗ f ∗ uσ˘ , \noindentsigma-breveAs r to usual the power , the of 2 =$ minus\breve 4 *{\ fsigma * u subr } breve-sigma − $ diameter comma of a cycles i s two t imes it s radius . Lemma 5 . 22 . \quad The $ \breve{\sigma} − $ radius of a cycle $Cˆ{ s } {\breve{\sigma}}$ i s equal to wherewhere sigma-breveσ˘r i s r the to the square power of of cycle 2 i s the ’ s squareσ˘− radius of cycle quoteright, uσ˘ i s the s breve-sigma second coordinate hyphen radius of its commaσ˘− focus u sub sigma-breve i s the second coordinate$1and / off kitsit breve-sigma s ,$ where hyphen $k$ focus and is(2,1) f it s − entry o f det − normalised focalfocal length length period . \noindentAn intuitiveAnmatrix intuitive notion of( notion a see distance Definition of ain distance both mathematics 3 in . 9both ) \ and mathematicsquad theof everyday the and cycle life the is usually .everyday of life is usually of a a variationalvariational nature nature period . .. We natural natural p perceive erceive the the shortest shortest distance distance b etween b etween two points two delivered points delivered \noindentby theby thestraightGeometrically straight lines and lines only and in then only various can then define canEPH it for define cases curves it this through for curves corresp an approximation through onds an t approximation o period the .. following This . This (variational evariational , h ) nature\quad nature echoesThe echoes valuealso in the also of following ( in 5 the . 1 definitionfollowing ) i s period the definition usual . radius of a circle or hyperbola ; \ hspace ∗{\ f i l l }( p ) \quad The diameter of a parabola i s the ( Euclidean ) distance b etween it s ( real ) roots , i . e . \quad s o l u −

\ hspace ∗{\ f i l l } tions of $ ku ˆ{ 2 } − 2 lu + m = 0 ,$ orrootsofits \quad ‘‘ adj oint ’’ \quad parabola $ − ku ˆ{ 2 } + 2 lu + m − \ f r a c { 2 l ˆ{ 2 }}{ k } = 0 $

\noindent ( see Fig . 10 (a) ) . Remark 5 . 3 . \quad Note that

\ [ \breve{\sigma} { r }ˆ{ 2 } = − 4 ∗ f ∗ u {\breve{\sigma}} , \ ]

\noindent where $ \breve{\sigma} { r }ˆ{ 2 }$ i s the square of cycle ’ s $ \breve{\sigma} − $ r a d i u s $ , u {\breve{\sigma}}$ i s the second coordinate of its $ \breve{\sigma} − $ focus and $ f $ i t s

\noindent focal length .

An intuitive notion of a distance in both mathematics and the everyday life is usually of a variational nature . \quad We natural p erceive the shortest distance b etween two points delivered by the straight lines and only then can define it for curves through an approximation . \quad This variational nature echoes also in the following definition . Erlangen Program at Large hyphen 1 : .... Geometry of Invariants .... 25 \noindenthlineErlangenErlangen Program Program at Large at - Large 1 :− 1 : \ h f i l l GeometryGeometry of of Invariants Invariants \ h f i l l 25 25 Definition 5 period 4 period .... The open parenthesis sigma comma sigma-breve closing parenthesis hyphen distance between two points is the\ [ extremum\ r u l e {3em of}{ breve-sigma0.4 pt }\ hyphen] diameters for all sigma hyphen cycles passing through both points period DuringDefinition geometry classes5 . 4 we . oftenly makeThe measurements(σ, σ˘)− distance with abetween compass comma two points which is is the based extremum on of σ˘− \noindentthediameters idea thatDefinition a cyclefor all is locus 5 . of 4 points . \ h equidistant f i l l The from $ ( its centre\sigma period We, can\breve expand{\ itsigma for all} cycles) − $ distance between two points is the extremum of $ \inbreve theσ− followingcycles{\sigma passing definition} − through$ : diameters both points for .all DefinitionDuring 5 period geometry 5 period classes .... The we sigma-breve oftenly make hyphen measurements length from a with sigma-ring a compass hyphen , centrewhich or is from based a ring-sigma on hyphen focus of a directed\noindentthe interval idea$ right that\sigma arrowa cycle minus-A-minus− is$ locus cycles of subpoints passing B i sequidistant through from both its points centre .. We can expand it for all thecycles passes to in the the power following of sigma-breve definition hyphen : radius of through the B period These to the power of sigma hyphen cycle with lengths its areDuring to theDefinition geometry power of sigma-ring 5 classes . 5 . sub we denoted oftenlyThe toσ˘− the makelength power measurements offrom hyphen a ˚σ centre− centre with or by or a to from compass the power a ˚σ− , offocus which ring-sigma of is a directed sub based l sub on c open parenthesis to the powerthe ofinterval idea minus-A-minus that→ minus a cycle sub− arrowright-BA − isminus locusB closingi of s points parenthesis equidistant to the power from of hyphen its focus centre correspondingly . We can and expand l sub f it open for parenthesis all cycles to the powerin the of minus-A-minus following definitionsub arrowright-B : closing parenthesis correspondingly period to the power of at the point A which Remarkσ˘− 5radius period 6 period σ−cycle ˚σ −centre ˚σ−focus @thepoint \noindentthepassesDefinitionofthroughthe 5B. . 5These . withlengthsits\ h f i l l Thearedenoted $ \breveorby{\sigmaminus−}A−minus − arrowright$ length−B correspondingly from a $ \ mathringminus−A−{\minussigmaarrowright}− −B $ Awhich 1 period .. Note that the distance is a symmetric functions of twolc points( by it s definition and) this andlf ( )correspondingly. centreis not necessarilyor from a true $ for\ mathring lengths period{\sigma .. For} modal − $ logic focus of non hyphenof a directed symmetric distancesinterval see comma$\rightarrow .. for { minus−A−minus } { B }$ i sexampleRemark comma 5 .. . open 6 . square bracket 5 4 closing square bracket period .. However the first .. axiom .. open parenthesis l open parenthesis x comma y1 closing . Note parenthesis that the = 0 distance iff x = y closing is a symmetric parenthesis functions .. should b of e modified two points as by it s definition and \ [Line thethis 1 follows{ isp nota s s: e Linenecessarily s }ˆ 2{\ openbreve parenthesis true{\ forsigma lengths l open} −parenthesis . Forr a d i modal u x s comma} o logic f { ythrough closing of non parenthesis -} symmetricthe ={ 0B distancesand l . open These see parenthesis , }ˆ{\ x commasigma y closing− c y c l e } with { l e n g t h s } parenthesisi t s {forare example= 0} closingˆ{\ mathring , parenthesis [ 5 4 ]{\ .sigma iff x However =}} y periodˆ{ −the firstc e n t r e axiom} { denoted(l(x, y)} =or { 0byiff }xˆ{\= mathringy) should{\sigma}}ˆ{ − f o c u s } { l { c } ( ˆ2{ periodbminus e modified ..− AA− cycleminus as i s uniquely{ arrowright defined by−B elliptic}} ) or} hyperboliccorrespondingly centre and a point{ and which it l passes{ f period} ( ˆ{ minus−A−minus { arrowright −B }} )However correspondingly the parabolic centre . } isˆ not{ @ so useful the period point .. Correspondingly} A which open\ ] parenthesis sigma comma 0 closing parenthesis hyphen length : from parabolic centre i s not properly defined period(l(x, y) = 0 and l(x, y) = 0) iff x = y. \noindent Remark 5 . 6 . Lemma 5 period2 . 7 periodA cycle i s uniquely defined by elliptic or hyperbolic centre and a point which it 1 periodpasses .. The. cycle of the form open parenthesis 3 period 1 1 closing parenthesis has zero radius period 1 .2 period\quad ..Note The .. that distance the .. distancebetween two is points a symmetric y = e sub 0 functionsu plus e sub 1 of v .. two and points y to the bypower it of s prime definition = e sub 0 and u to the this power of is not necessarilyHowever the true parabolic for lengths centre is . not\quad so usefulFor modal . Correspondingly logic of non − (symmetricσ, 0)− length distances from see , \quad f o r prime plusparabolic e sub 1 v to the power of prime .. in the .. e l liptic .. or examplehyperbolic , spaces\quad is [ 5 4 ] . \quad However the first \quad axiom \quad $(l(x,y)=0$ i f f $ x = y ) $ \quad shouldcentre i b s enot modified properly as defined . Equation:Lemma open 5 parenthesis . 7 . 5 period 2 closing parenthesis .. d to the power of 2 open parenthesis y comma y to the power of prime closing parenthesis = sigma-breve open parenthesis open parenthesis u minus u to the power of prime closing parenthesis to the power of 2 minus sigma \ [ \ begin { a l i g n e d }\1succ . The: \\ cycle of the form ( 3 . 1 1 ) has zero radius . open parenthesis v minus v to the power of prime closing parenthesis to the power of 2 closing0 parenthesis0 plus 40 open parenthesis 1 minus ( l2 . (x,y)=0andl The distance between two points (x,y)=0)y = e0u + e1v and y = e0u iff+ e1v x=y. \end{ a l i g n e d }\ ] sigma sigma-brevein the eclosing l liptic parenthesis or vv to the power of prime divided by open parenthesis u minus u to the power of prime closing parenthesis to the power of 2 breve-sigma minus open parenthesishyperbolic v minus spaces v to the is power of prime closing parenthesis to the power of 2 parenleftbig open parenthesis u minus u to the power of prime closing parenthesis to the power of 2 minus sigma open parenthesis v minus v to the power of prime closing\ hspace parenthesis∗{\ f i l l } to2 the . power\quad ofA 2 parenrightbigcycle i s uniquely comma defined by elliptic or hyperbolic centre and a point which it passes . and in parabolic case it is openσ˘ parenthesis((u − u0)2 − seeσ( Figv − periodv0)2) + 1 4(1 0 open− σσ˘ parenthesis)vv0 b closing parenthesis .. and open square bracket 7 2 comma d2(y, y0) = ((u − u0)2 − σ(v − v0)2), (5.2) p\ hspace period 38∗{\ commaf i l l } openHowever parenthesis the parabolic5 closing(u parenthesis− u0 centre)2σ˘ − (v closing− isv0) not2 square so bracket useful closing . \quad parenthesisCorrespondingly $ ( \sigma , 0Equation: ) − $ open length parenthesis from 5 period parabolic 3 closing parenthesis .. d to the power of 2 open parenthesis y comma y to the power of prime closing parenthesis = openand inparenthesis parabolic u minus case it u tois the( see power Fig of . prime1 0 closing ( b ) parenthesisand [ 7 to 2 the, p power. 38 , of ( 52 period ) ] ) \ centerlineProof period{ centre .. Let C i sub s s not to the properly power of defined sigma open . parenthesis} l closing parenthesis b e the family of cycles passing through both points open parenthesis u comma v closing parenthesis and open parenthesis u to the power of prime comma v to the power of prime closing 2 0 0 2 parenthesis\noindent openLemma parenthesis 5 . 7 under . d (y, y ) = (u − u ) . (5.3) the assumption v equal-negationslash v to the power of prime closing parenthesis and parametrised by its coefficient l in the defining equation Proof . Let Cσ(l) b e the family of cycles passing through both points (u, v) and (u0, v0)( open\ centerline parenthesis{1 2 period . \quad 1 0sThe closing cycle parenthesis of the period form .. By ( 3 . 1 1 ) has zero radius . } under the assumption v 6= v0) and parametrised by its coefficient l in the defining equation ( a calculation2 . 1 0 ) done . in By G i aN calculationaC open square done bracket in G 4 i 6 N comma aC [ S 4 3 6 period , §3 5. period 5 . 1 1] weclosing found square that bracket the only we found that the only critical point\ hspace of det∗{\ openf i l lparenthesis}2 . \quad C subThe s to\ thequad powerd i s of t a sigma n c e \ openquad parenthesisbetween l closingtwo points parenthesis $ yclosing = parenthesis e { 0 i} s : u + e { 1 } critical point of det (Cσ(l)) i s : v $Equation:\quad openand parenthesis $ y ˆ{\ 5prime periods 4} closing= parenthesis e { 0 } .. lu sub ˆ{\ 0 =prime 1 divided} by+ 2 parenleftbigg e { 1 } openv ˆ parenthesis{\prime u} to$ the\quad power ofin prime the \quad e l l i p t i c \quad or plus u closing parenthesis plus open parenthesis sigma-breve sigma minus 1 closing parenthesis open parenthesis u to the power of prime minus \ centerline { hyperbolic spaces is } u closing parenthesis open parenthesis v to1 the power of 2 minus v(u to0 − theu)( powerv2 − v of02) prime 2 closing parenthesis divided by open parenthesis u to l = ((u0 + u) + (˘σσ − 1) ), (5.4) the power of prime minus u closing parenthesis0 2 to the power of( 2u0 breve-sigma− u)2σ˘ − (v minus− v0)2 open parenthesis v minus v to the power of prime closing parenthesis\ begin { a l to i g then ∗} power of 2 parenrightbigg comma d ˆ{ 2 } ( y , y ˆ{\prime } ) = \ f r a c {\breve{\sigma} ( ( u − u ˆ{\prime } ) ˆ{ 2 } open( Note parenthesis that Notein the that case in theσσ˘ case= 1, sigmai . e . sigma-breve both points = 1 and comma cycles i period spaces e period are simultaneously both points and either cycles spaces are simultaneously − \sigma ( v − v ˆ{\prime } ) ˆ{ 2 } ) + 4 ( 1 − \sigma \breve{\sigma} ) vv ˆ{\prime }}{ ( either ellipticelliptic or hyperbolic , this expression reduces t o the expected midpoint l = 1 (u+u0).) Since u − u ˆ{\prime } ) ˆ{ 2 }\breve{\sigma} − ( v − v0 ˆ{\2 prime } ) ˆ{ 2 }} ( ( u − or hyperbolicin the elliptic comma or this hyperbolic expression reduces case the t o parameter the expectedl midpointcan take l sub any 0 real = 1 divided value , by the 2 open extremum parenthesis of u plus u to the power of u ˆ{\prime } ) ˆ{ 2 } − \sigma ( v − v ˆ{\prime } ) ˆ{ 2 } ), \ tag ∗{$ ( 5 . 2 prime closingdet (C parenthesisσ(l)) period closing parenthesis .. Since in the ) $} s elliptici s reached or hyperbolic in l0 caseand the is parameter equal to ( l can5 . 2take ) ( any calculated real value by commaG i N the aC extremum[ 4 6 , § of3 . det 5 .open 1 ] )parenthesis . A C sub s to the power of sigma\end{ openseparatea l i g parenthesisn ∗} calculation l closing parenthesis closing parenthesis 1 0 1 i s reached in l sub 0 and0 is equal to open parenthesis 5 period 2 closingparabolic parenthesis opencase parenthesis1 possible calculatedvalues of bylare Geither i Nin( aC−∞ open, 2 (u square+u )), or ( (u+ 0 for the case v = v gives the same answer . or Inthetheonly value i s l = the 0 2 u ), ∞), upward bracket\ centerline 4 6 comma{and S 3 in period parabolic 5 period 1case closing it square is ( bracket see Fig closing . 1 parenthesis 0 ( b ) period\quad ....and[72,p.38,(5)]) A separate2 (u calculation+u )sinceforthatvalue aparabolashould} flipbetween forand the case downward v = v to directions the power of of prime its branches gives the same . In answer any period of those cases the extremum value corresp \ begin { a l i g n ∗} l = 1 (u + u0) (5.3). or Inonds the the t o only the toboundary the power point of parabolic2 value iand s to the i s powerequal of to case l = the 1 divided by 2 sub open parenthesis u to the power of possibled ˆ{ 2 sub} plus( u to y the , power y of ˆ{\ primeprime closing} parenthesis) = since ( to u the− poweru of ˆ{\ valuesprime sub for} to) the ˆ{ power2 } of. of\ subtag ∗{ that$ to ( the 5 power . of l3 are sub ) $ value} to the power of either sub a parabola to the power of in open parenthesis minus infinity comma 1 divided by 2 sub should to the\end power{ a l ig of n open∗} parenthesis u to the power of plus u to the power of prime sub flip to the power of closing parenthesis closing parenthesis comma to the power of or sub b etween to the power of open parenthesis 1 divided by 2 open parenthesis u plus u to the power of prime closing parenthesis\noindent commaProof infinity . \quad closingLet parenthesis $ C ˆ{\ commasigma upward} { s } ( l ) $ b e the family of cycles passing through both points $(and downward u , directions v )$and$( of its branches period uˆ{\ .. Inprime any of} those, cases v the ˆ{\ extremumprime } value) corresp ( $ onds under thet o the assumption boundary point $ v l = 1\ dividedne byv ˆ2{\ openprime parenthesis} ) u $ plus and u to the parametrised power of prime by closing its parenthesis coefficient and i s equal $ l $to open in parenthesis the defining equation ( 2 . 1 0 ) . \quad By 5a period calculation 3 closing parenthesis done in period G i NaC blacksquare [ 4 6 , \S 3 . 5 . 1 ] we found that the only critical point of det $ ( C ˆ{\sigma } { s } ( l ) ) $ i s :

\ begin { a l i g n ∗} l { 0 } = \ f r a c { 1 }{ 2 } ( ( u ˆ{\prime } + u ) + ( \breve{\sigma}\sigma − 1 ) \ f r a c { ( u ˆ{\prime } − u ) ( v ˆ{ 2 } − v ˆ{\prime 2 } ) }{ ( u ˆ{\prime } − u ) ˆ{ 2 }\breve{\sigma} − ( v − v ˆ{\prime } ) ˆ{ 2 }} ), \ tag ∗{$ ( 5 . 4 ) $} \end{ a l i g n ∗}

\noindent ( Note that in the case $ \sigma \breve{\sigma} = 1 , $ i . e . both points and cycles spaces are simultaneously either elliptic or hyperbolic , this expression reduces t o the expected midpoint $ l { 0 } = \ f r a c { 1 }{ 2 } ( u + u ˆ{\prime } ) . ) $ \quad Since in the elliptic or hyperbolic case the parameter $ l $ can take any real value , the extremum of det $ ( C ˆ{\sigma } { s } ( l ) ) $

\noindent i s reached in $ l { 0 }$ and is equal to ( 5 . 2 ) ( calculated byG i NaC [ 4 6 , \S 3 . 5 . 1 ] ) . \ h f i l l A separate calculation

\noindent for thecase $v = vˆ{\prime }$ gives the same answer . or $ In the { the } only ˆ{ p a r a b o l i c }$ value i $ s ˆ{ case } l = the \ f r a c { 1 }{ 2 } { ( u }ˆ{ p o s s i b l e } { + u ˆ{\prime } ) s i n c e }ˆ{ values } { f o r }ˆ{ o f } { that }ˆ{ l are } { value }ˆ{ e i t h e r } { a parabola }ˆ{ in ( − \ infty , \ f r a c { 1 }{ 2 }}ˆ{ ( u } { should }ˆ{ + u ˆ{\prime }}ˆ{ )) , } { f l i p }ˆ{ or } { b etween }ˆ{ ( \ f r a c { 1 }{ 2 } ( u + } u ˆ{\prime } ), \ infty ) , { upward }$ and downward directions of its branches . \quad In any of those cases the extremum value corresp onds t o the boundary point $ l = \ f r a c { 1 }{ 2 } ( u + u ˆ{\prime } ) $ and i s equal to $ ( 5 . 3 ) . \ blacksquare $ 26 .... V period V period Kisil \noindenthline26 26 \ h f i l l V . V . K i s i l V . V . Kisil Corollary 5 period 8 period .. If cycles C sub sigma-breve to the power of s .. and tilde-C sub sigma-breve to the power of s .. are normalised by\ [ conditions\ r u l e {3em k =}{ 10.4 .. and pt }\ tilde-k] = 1 then angbracketleft C sub sigma-breve to the power of comma to the power of s tilde-C sub sigma-breve to the power of s right angbracket = bar s ˜s c minusCorollary tilde-c bar sub 5 . sigma-breve 8 . If to cycles the powerCσ˘ of 2and minusC breve-sigmaσ˘ are normalised r to the power by conditions of 2 minus breve-sigmak = 1 and r-tilde to the power of 2 sub ˜ comma\noindentk = 1 thenCorollary 5 . 8 . \quad If cycles $Cˆ{ s } {\breve{\sigma}}$ \quad and $ \ tilde {C} ˆ{ s } {\breve{\sigma}}$ \quadwhereare bar normalised c minus c bar subby sigma-breveconditions to the $ k power = of 2 1 = $ open\quad parenthesisand l $ minus\ tilde l-tilde{k} closing= parenthesis 1 $ then to the power of 2 minus s ˜s 2 2 2 sigma-breve open parenthesis n minus tilde-n closinghCσ˘, Cσ˘ parenthesisi =| c − c˜ |σ˘ to− theσ˘r − powerσ˘r˜, of 2 is the square of sigma-breve hyphen distance between cycles \ [ \ langle C ˆ{ s } {\breve{\sigma} ˆ{ , }}\ tilde {C} ˆ{ s } {\breve{\sigma}}\rangle = \mid quoteright centres comma2 r sub˜ breve-sigma2 and2 r-tilde sub breve-sigma where | c − c | = (l − l) − σ˘(n − n˜) is the square of σ˘− distance between cycles ’ centres , rσ˘ c are− bs hyphen \ tilde radii{c}\ ofσ˘ themid respectiveˆ{ 2 cycles} {\ periodbreve{\sigma} } − \breve{\sigma} { r }ˆ{ 2 } − \breve{\sigma} {\ tilde { r }}ˆ{ 2 } { , }\ ] and r˜σ˘ To getare feelingbs− ofradii the ofidentity the respective open parenthesis cycles 5 . period 2 closing parenthesis we may observe comma that : d to the power of 2 open parenthesis y comma y to the power of prime closing parenthesis = open parenthesis u minus u to the power of \noindentTo getwhere feeling $ of\ themid identityc − ( 5 .c 2 ) we\mid mayˆ{ observe2 } {\ , thatbreve : {\sigma}} = ( l − \ tilde { l } ) ˆ{ 2 } prime closing parenthesis to2 the power0 of 20 plus2 open parenthesis0 2 v minus v to the power of prime closing parenthesis to the power of 2 comma − \breve{\sigma} d ((y, y ) n = (u−− u ) \+tilde (v − {vn)}, for) ˆ elliptic{ 2 }$ values is theσ square=σ ˘ = −1 of, $ \breve{\sigma} − $ distance between cycles ’ centres .. for elliptic values .. sigma2 =0 sigma-breve0 2 = minus 10 comma2 $ , r {\breve{\dsigma(y, y )}} = (u$− u and) − ( $v −\ tildev ) , {forr } hyperbolic{\breve{\ valuessigma}}σ =$ σ ˘ = 1, d toi .the e .power these of are 2 open familiar parenthesis expressions y comma for y the to the elliptic power and of prime hyperbolic closing parenthesis spaces . However = open parenthesis four other u minus u to the power of prime closing parenthesis to the power of 2 minus open parenthesis v minus v to the power of prime closing parenthesis to the power of 2 comma \noindentcases are $ bs − $ radii of the respective cycles . .. for hyperbolic(σσ˘ = −1 valuesor 0 ) .. gives sigma quite = sigma-breve different = results 1 comma . For example , d2(y, y0)arrowright − negationslash0 if i period e period these0 are familiar expressions for the elliptic and hyperbolic spaces period However four other cases \noindenty tenseTo to gety in feeling the of the identity ( 5 . 2 ) we may observe , that : openusual parenthesis sense .sigma sigma-breve = minus 1 or 0 closing parenthesis gives quite different results period .... For example comma d to the power of 2 open parenthesis y comma y to the power of prime closing parenthesis arrowright-negationslash 0 if y tense to y to the power of \ centerlineRemark{ $ 5 d . 9ˆ{ . 2 } ( y , y ˆ{\prime } ) = ( u − u ˆ{\prime } ) ˆ{ 2 } + ( prime in the 1 . In the three cases σ =σ ˘ = −1, 0 or 1 , which were typically studied b efore , v usual− sensev ˆ{\ periodprime } ) ˆ{ 2 } , $ \quad for elliptic values \quad $ \sigma = \breve{\sigma} = − 1the ,above $ } Remark 5 perioddistances 9 period are conveniently defined through the Clifford algebra multiplications 1 period .. In the three cases sigma = sigma-breve = minus 1 comma 0 or 1 comma .. which were typically studied b efore comma .. the \ centerline { $ d ˆ{ 2 } ( y , y ˆ{\prime } ) = ( u − u ˆ{\prime } ) ˆ{ 2 } − ( above 2 2 d (ue0 + ve1) = −(ue0 + ve1) . v distances− v are ˆ{\ convenientlyprime } defined) ˆ{ through2 } ,e,p,h the $ Clifford\quad algebrafor hyperbolic multiplications values \quad $ \sigma = \breve{\sigma} =d sub1 e comma,2 $ .} p Unlesscomma hσ to= theσ ˘ the power parabolic of 2 open distance parenthesis ( 5 ue . sub 3 ) 0 i plus s not ve received sub 1 closing from parenthesis ( 5 . 2 ) = by minus the open parenthesis ue sub 0 plus vesubstitution sub 1 closing parenthesis to the power of 2 period \noindent2 period ..i Unless . e . sigma these = sigma-breve are familiar the parabolic expressions distance foropen the parenthesis elliptic 5 period and 3 hyperbolic closing parenthesis spaces i s not . Howeverreceived from four open other cases parenthesis 5 period 2 closing parenthesis by the substitutionσ = 0. \noindentsigma = 0 period$ ( \sigma \breve{\sigma} = − 1 $ or 0 ) gives quite different results . \ h f i l l For example $ , d ˆ{ 2 } ( y ,Now y we ˆ{\ turnprime to calculations} ) arrowright of the lengths−negationslash . 0$ if $y$ tense to $y ˆ{\prime }$ NowLemma we turn to 5 calculations . 10 . of the lengths period inLemma the 5 period 10 period 1 . The σ˘− length from the ˚σ− centre between two points y = e u + e v and y0 = e u0 + e v0 is 1 period .. The sigma-breve hyphen length from the sigma-ring hyphen centre between0 1 two points y = e0 sub 01 u plus e sub 1 v and y to the power\noindent of primeusual = e sub sense 0 u to . the power of prime plus e sub 1 v to the power of prime is Equation: open parenthesis 5 period 52 closing0 parenthesis0 2 .. l0 sub2 c sub0 sigma-breve2 to the power of 2 open parenthesis y comma y to the lc (y, y ) = (u − u ) − σv + 2˚σvv − σv˘ . (5.5) power\noindent of primeRemark closing parenthesis5 . 9 . = openσ˘ parenthesis u minus u to the power of prime closing parenthesis to the power of 2 minus sigma v to 0 0 0 the2 power . of The primeσ˘ 2− pluslength 2 sigma-ring from the vv to˚σ− thefocus power between of prime two minus points sigma-brevey = e0 vu to+ e the1v and power ofy 2= periode0u + e1v is \ hspace2 period∗{\ ..f The i l l } sigma-breve1 . \quad hyphenIn the length three from thecases sigma-ring $ \sigma hyphen focus= between\breve two{\sigma points} y == e sub− 0 u plus1 e sub, 1 0 v $and or y to 1 the , \quad which were typically studied b efore , \quad the above power of prime = e sub 0 u to the power of prime plus e sub 1 v to the power of prime is \ centerline { distances are convenientlyl2 (y, ydefined0) = (˚σ − σ˘ through)p2 − 2vp, the Clifford algebra multiplications(5.6) } Equation: open parenthesis 5 period 6 closingfσ˘ parenthesis .. l sub f sub sigma-breve to the power of 2 open parenthesis y comma y to the power of prime closing parenthesis = open parenthesis sigma-ring minus sigma-breve closing parenthesis p to the power of 2 minus 2 vp comma \ [where d ˆ{ 2 } { e , p , h } ( ue where{ 0 } + ve { 1 } ) = − ( ue { 0 } + ve { 1 } ) ˆEquation:{ 2 } . open\ ] parenthesis 5 period 7 closing parenthesis .. p = ring-sigma open parenthesis minus open parenthesis v to the power of prime minus v closing parenthesis plusminux radicalbig-line of sigma-ring open parenthesis u to the power of prime minus u closing parenthesis to the 0 p 0 2 0 2 02 power of 2 plus open parenthesisp = ˚σ v(− to(v the− powerv) ± of˚σ( primeu − u minus) + (v v− closingv) − parenthesisσ˚σv ), if to˚σ the6= power0, of 2 minus sigma(5.7) ring-sigma v to the power \ hspace ∗{\ f i l l }2 . \quad Unless $ \sigma = \breve{\sigma} $ the parabolic distance ( 5 . 3 ) i s not received from ( 5 . 2 ) by the substitution of prime 2 closing parenthesis comma if sigma-ring negationslash-equal(u0 − u 0)2 comma− σv02 Equation: open parenthesis 5 period 8 closing parenthesis .. p p = if ˚σ = 0. (5.8) = open parenthesis u to the power of prime minus u closing parenthesis0 to the power of 2 minus sigma v to the power of prime 2 divided by 2 2(v − v) , open\ [ \ parenthesissigma = v to 0the power . \ ] of prime minus v closing parenthesis sub comma if ring-sigma = 0 period ProofProof period . .. IdentityIdentity open ( parenthesis 5 . 5 ) is verified 5 period in5 closingG i NaC parenthesis[ 4 6 , § is3 verified . 5 . 4 in ] .G i NaC For open the second square bracket part 4 6 comma S 3 period 5 period 4 closing square bracket period .. For the second part we observe that the 0 0 \ centerlinewe observe{Now that we turn the parabola to calculations with the focus of the(u, lengths v) passing . through} (u , v ) has the following parabolaparameters with the : focus open parenthesis u comma v closing parenthesis passing through open parenthesis u to the power of prime comma v to the power of prime closing parenthesis has the following parameters : \noindent Lemma 5 . 10 . k = 1 comma l = u comma nk == p 1, comma l = u, m = n = 2 ring-sigmap, m = 2˚σpv pv0 to− theu02 + power 2uu0 of+ σv prime02. minus u to the power of prime 2 plus 2 uu to the power of prime plus sigma v to the power of prime 2 period \ centerlineThenThen the formula the{1 formula . open\quad parenthesis (The 5 . 6 ) $ i 5\ s periodbreve verified{\ 6 closingsigma by the} parenthesisG i− NaC$ i lengthcalculation s verified from by the [ 4 the G 6 ,i NaC§ $3.5\. calculationmathring5].  {\ opensigma square} bracket − $ 4 centre 6 comma between S two points 3$ period yRemark = 5 period e { 5 5 closing0 . 1} 1 squareu . + bracket e { period1 } blacksquarev $ and $ y ˆ{\prime } = e { 0 } u ˆ{\prime } + e { 1 } v ˆRemark{\prime1 5 . period} The$ 1 i 1value s period} of p in ( 5 . 7 ) i s the focal length of either of the two cycles , which 1 periodare in .. theThe value of p in open parenthesis 5 period 7 closing parenthesis .. i s the focal length of either of the two cycles comma .. which are\ begin in the{ aparabolic l i g n ∗} case upward or downward parabolas ( corresponding t o the plus or minus signs lparabolic ˆ{) 2 } case{ c upward{\breve or downward{\sigma parabolas}}} ( open y parenthesis , y corresponding ˆ{\prime } t o the) plus = or (minus u signs− closingu ˆ parenthesis{\prime } ) ˆ{ 2 } 0 0 −with \sigma focus at openv ˆ{\ parenthesisprime with u2 comma focus} + v at closing 2(u, v parenthesis)\ mathringand passing and{\sigma throughpassing} through(vvu , v ˆ){\ open. prime parenthesis} − u to \ thebreve power{\sigma of prime} commav ˆ{ v2 to} the. \ powertag ∗{ of$ prime ( 5 closing . parenthesis 5 ) $ period} \end{ a l i g n ∗}

\ centerline {2 . \quad The $ \breve{\sigma} − $ length from the $ \ mathring {\sigma} − $ focus between two points $ y = e { 0 } u + e { 1 } v $ and $ y ˆ{\prime } = e { 0 } u ˆ{\prime } + e { 1 } v ˆ{\prime }$ i s }

\ begin { a l i g n ∗} l ˆ{ 2 } { f {\breve{\sigma}}} ( y , y ˆ{\prime } ) = ( \ mathring {\sigma} − \breve{\sigma} ) p ˆ{ 2 } − 2 vp , \ tag ∗{$ ( 5 . 6 ) $} \end{ a l i g n ∗}

\ centerline {where }

\ begin { a l i g n ∗} p = \ mathring {\sigma} ( − ( v ˆ{\prime } − v ) \pm \ sqrt {\ mathring {\sigma} ( u ˆ{\prime } − u ) ˆ{ 2 } + ( v ˆ{\prime } − v ) ˆ{ 2 } − \sigma \ mathring {\sigma} v ˆ{\prime 2 }} ) , i f \ mathring {\sigma}\not= 0 , \ tag ∗{$ ( 5 . 7 ) $}\\ p = \ f r a c { ( u ˆ{\prime } − u ) ˆ{ 2 } − \sigma v ˆ{\prime 2 }}{ 2 ( v ˆ{\prime } − v ) } { , } i f \ mathring {\sigma} = 0 . \ tag ∗{$ ( 5 . 8 ) $} \end{ a l i g n ∗}

\noindent Proof . \quad Identity ( 5 . 5 ) is verified in G i NaC [ 4 6 , \S 3 . 5 . 4 ] . \quad For the second part we observe that the parabola with the focus $ ( u , v ) $ passing through $ ( uˆ{\prime } , v ˆ{\prime } ) $ has the following parameters :

\ [k=1,l=u,n=p,m=2 \ mathring {\sigma} pv ˆ{\prime } − u ˆ{\prime 2 } + 2 uu ˆ{\prime } + \sigma v ˆ{\prime 2 } . \ ]

\noindent Then the formula ( 5 . 6 ) i s verified by the G i NaC calculation [ 4 6 , \S $ 3 . 5 . 5 ] . \ blacksquare $

\noindent Remark 5 . 1 1 .

\ hspace ∗{\ f i l l }1 . \quad Thevalue of $p$ in (5 . 7) \quad i s the focal length of either of the two cycles , \quad which are in the

\ hspace ∗{\ f i l l } parabolic case upward or downward parabolas ( corresponding t o the plus or minus signs )

\ centerline { with focus at $( u , v )$ andpassing through $( uˆ{\prime } , v ˆ{\prime } ) . $ } Erlangen Program at Large hyphen 1 : .... Geometry of Invariants .... 2 7 \noindenthlineErlangenErlangen Program Program at Large at - Large 1 :− 1 : \ h f i l l GeometryGeometry of of Invariants Invariants \ h f i l l 2 7 2 7 2 period .. In the case sigma sigma-breve = 1 the length open parenthesis 5 period 5 closing parenthesis b ecame the standard elliptic or hyperbolic\ [ \ r u l e { distance3em}{0.4 pt }\ ] open parenthesis u minus u to the power of prime closing parenthesis to the power of 2 minus sigma open parenthesis v minus v to the power of prime closing2 . parenthesis In the case to theσ powerσ˘ = 1 the of 2 length obtained (in 5 . open 5 ) parenthesisb ecame the 5 period standard 2 closing elliptic parenthesis or hyperbolic period .. Since these expressions \ hspacedistance∗{\ f i l l }2 . \quad In the case $ \sigma \breve{\sigma} = 1 $ the length ( 5 . 5 ) b ecame the standard elliptic or hyperbolic distance appeared both as0 distances2 0 2 and lengths(u − u they) − areσ( widelyv − v ) usedobtained period .. in On ( the 5 . other 2 ) hand . in Since the parabolic these expressions space we get threeappeared both as $additional (distances u lengths− andu b ˆ esides lengths{\prime of distance they} are) open ˆwidely{ parenthesis2 } used − . 5 \ periodsigma On the3 closing( other parenthesis v hand− inv the ˆ{\ parabolicprime } space) ˆ we{ 2 }$ obtained in ( 5 . 2 ) . \quad Since these expressions appeared both as distances andl subget lengths c sub three sigma-breve they additional are to the widely lengths power used of b 2 esides open . \ parenthesisquad of distanceOnthe y comma ( 5 other . 3 y ) to hand the power in the of prime parabolic closing parenthesis space we = open get parenthesis three u minus uadditional to the power of lengths prime closing b esides parenthesis of to distance the power ( of 5 2 plus . 3 2 ) vv to the power of prime minus breve-sigma v to the power of 2 l2 (y, y0) = (u − u0)2 + 2vv0 − σv˘ 2 parametrised by sigma-breve open parenthesiscσ˘ cf period Remark 1 period 1 period 1 closing parenthesis period \ [3 l period ˆ{ 2 ..} The{ c parabolic{\breve distance{\sigma open parenthesis}}} ( 5 periody , 3 closing y ˆ{\ parenthesisprime } can) b e expressed = ( as u − u ˆ{\prime } ) ˆ{ 2 } + 2parametrised vv ˆ{\prime by σ˘}( cf − . Remark \breve 1{\ .sigma 1 . 1} ) .v 3 . ˆ{ 2 The}\ ] parabolic distance ( 5 . 3 ) can b e d toexpressed the power asof 2 open parenthesis y comma y to the power of prime closing parenthesis = p to the power of 2 plus 2 open parenthesis v minus v to the power of prime closing parenthesis p in terms of the focal length open parenthesisd 52( periody, y0) = 7p closing2 + 2(v parenthesis− v0)p comma which i s an expression similar t o open parenthesis 5 period\noindent 6 closingparametrised parenthesis period by $ \breve{\sigma} ($ cf . Remark1 . 1 . 1) . 35 . period\quadin 2 ..The terms Conformal parabolic of the properties focal distance length of M dieresis-o ( ( 5 .5 7 . ) bius 3 , which ) maps can i b s an e expressed expression similaras t o ( 5 . 6 ) . All5 l open . 2 parenthesis Conformal to the power properties of minus-A-minus of M subo¨ bius B-arrowright maps sub closing parenthesis to the power of lengths = c l are to the power of\ [ open d ˆ parenthesis{ 2 } ( to the y power , of y minus-A-minus ˆ{\prime } sub) corresp = to p the ˆ{ power2 } of+ B-arrowright 2 ( closing v − parenthesisv ˆ{\ subprime onding} to the) power p \ ] of in R to the power of sigma to the power of fromminus Definition−A−minus cycles : 5 periodσ circles sub comma to the power of 5 are such parabolas to the power lengths ( B−arrowright)inR from 5 that all curves All minus−A−minus = cl Definitioncycles : 5.circles aresuchparabolas orforafixedhyperbolas, point A level objofects of that orl for( a fixed hyperbolasB−arrowright comma) pointare which Acorresp are to the poweronding of all level covariant to the power, of curves obj of ects which are covariant \ centerlinein the appropriate{ in terms geometries of the period focal .... Thus length we can ( expect 5 . 7 some ) , covariant which properties i s an expression of distances and similar t o ( 5 . 6 ) . } lengthsin the period appropriate geometries . Thus we can expect some covariant properties of distances \noindentDefinitionand 5 .. .5 period 2 \quad 1 2Conformal period .. We properties say that .. a distance of M $ or\ addot length{o} d i$ s SL bius sub maps2 open parenthesis R closing parenthesis hyphen conformallengths if for .fixedDefinition y comma 5 . 1 2 . We say that a distance or a length d i s SL2(R)− \ [Line Allconformal 1 y{ tol the (powerif ˆ for{ minus of fixed prime−Ay,− inminus R to the{ powerB−arrowright of sigma the}}} limitˆ{ Linel e n 2 glimint t h s } t{ right) } arrow= 0 d c open l parenthesis{ are } gˆ{ times( ˆ y{ commaminus− gA−minus }}ˆ{ B−arrowright times) } { openc o r parenthesis r e s p }ˆ{ yin plus ty R to ˆ{\ thesigma power of}} prime{ onding closing parenthesis}ˆ{ from closing} D e f parenthesis i n i t i o n { dividedc y c l e s by} d open: parenthesis 5 . { c y i comma r c l e s y}ˆ plus{ 5 } { , } y0 ∈ σthelimit tyare to the such power{ ofparabolas prime closing} parenthesisˆ{ that } subor comma for where ga in SL fixed sub 2{ openRhyperbolas parenthesis R} closing, pointparenthesis{ commawhich open} A parenthesis{ are } 5ˆ{ a l l } 0 periodl e v e l 9 closing{ c o v parenthesis a r i a n t }ˆ{ curvesd(g} · y,obj g · (y + ty o f )) { e c t s }\ ] lim where g ∈ SL2( ), (5.9) t→0 0 R exists and its value depends only from yd( andy, y + g andty ) i s independent, from y to the power of prime period The following proposition shows that SL sub 2 open parenthesis R closing parenthesis hyphen conformality i s not rare period y g y0. \noindentPropositionexistsin and 5 period the its value appropriate 1 3 period depends geometries only from .and\ h f iand l l Thus i s independent we can expect from some covariant properties of distances and SL ( )− 1 period .. TheThe distance following .. open proposition parenthesis 5shows period that 2 closing2 parenthesisR conformality .. is conformal i s not if and rare only . if the type of point and cycle spaces are\noindent thePropositionl e n g t h s 5 . . 1 3 . Dsame e f i n i comma t i1 o n .\ ..quad i periodThe5 distance .e period 1 2 . sigma\quad( 5 sigma-breve .We 2 ) sayis = that conformal 1 period\quad .. if Thea and distanceparabolic only if distance the or type a ..length openof point parenthesis $d$ and cycle 5 i period s $SL 3 closing{ parenthesis2 } ( R).. is conformalspaces− $ are only conformal the in the parabolic if for point fixed $ y , $ .σσ˘ = 1. space periodsame , i . e The parabolic distance ( 5 . 3 ) is conformal only in the \ [ \2begin periodparabolic{ ..a l iThe g n point e lengths d } y ˆfrom{\prime centres open}\ parenthesisin R ˆ{\ 5 periodsigma 5 closing} the parenthesis l i m i t are\\ conformal for any combination of values of sigma comma\lim sigma-breve{ t \ andrightarrow sigma-ring sub0 period}\ f r a c { dspace ( . g \cdot y , g \cdot ( y + ty ˆ{\prime } ))3 period}{2d ... The ( Thelengths y lengths from , foci fromy .. +open centres parenthesisty ˆ({\ 5prime . 5period ) are} conformal 6) closing} { parenthesis, } forwhere any .. combination are g conformal\ in of for valuesSL ring-sigma{ of2 } negationslash-equal(R), 0 .. σ, σ˘ ˚σ ˚σ 6= 0 and( any 5 combinationand . 9. 3 of ) . values\end{ The ofa l sigma i g lengths n e d }\ ] from foci ( 5 . 6 ) are conformal for and any σ andcombination sigma-breve sub of period values of Proof period .... This i s another straightforward calculation in G i NaC open square bracket 4 6 comma S 3 period 5 period 2 closing square andσ˘ bracket\noindent periodexists blacksquare and its value depends only from. $ y $ and $ g $ and i s independent from $ y ˆ{\prime } . $ TheProof conformal . property of theThis distance i s another open parenthesis straightforward 5 period 2 calculation closing parenthesis in G i endash NaC open[ 4 6 parenthesis , § 3.5.2]. 5 period 3 closing parenthesis .. fromThe Proposition conformal 5 period property 1 3 period of the 1 i s distance well hyphen ( 5 known . 2 ) – comma ( 5 . .. 3 of) from Proposition 5 . 1 3 . 1 i s \ centerlinecoursewell comma - known{The .. see following , .. open of course square proposition bracket, see 1 6 comma [shows 1 6 , .. that 7 7 2 2 closing ] $ . SL square However{ 2 bracket} (R) periodthe .. same However− $ property .. conformalitythe .. same .. property i s not .. of rare non . } hyphenof symmetric non - symmetric .. lengths .. from lengths .. Proposi from hyphen Proposi - t ions 5 . 1 3 . 2 and 5 . 1 3 . 3 could \noindent Proposition 5 . 1 3 . t ionsb e .. hardly 5 period expected 1 3 period . 2 and The 5 period smaller 1 .. group3 periodSL 3 could2(R)( b ein hardly comparison expected period .. The smaller group SL sub 2 open parenthesis R closingt o parenthesis all linearopen - fractional parenthesis transforms in comparison of R2) generates bigger number of conformal metrics , cf . \ hspacet oRe all∗{\ linear - markf i hyphenl l } 31 . . 5fractional\ .quad The transforms distance of R to\quad the power( 5 of . 2 2 closing ) \quad parenthesisis conformal generates bigger if and number only of conformalif the type metrics of comma point and cycle spaces are the cf period ReThe hyphen exception of the case ˚σ = 0 from the conformality in 5 . 1 3 . 3 looks \ hspacemarkdisappointing 3∗{\ periodf i l l 5} periodsame on ,the\quad first glancei . e , $ especially. \sigma in the\breve light{\ ofsigma the parabolic} = Cayley 1 . transform $ \quad The parabolic distance \quad ( 5 . 3 ) \quad is conformal only in the parabolic point Theconsidered .. exception lat of the er case in .. Section ring-sigma 8 = . 0 from 2 . the conformality However a in detailed .. 5 period study 1 3 period of algebraic 3 .. looks structure .. disappointing on \ centerlinetheinvariant first glance{ space under comma . parabolic ..} especially rota in the - light t ions of the parabolic [ 4 8 , Cayley 4 7 ] transform removes considered obscurity lat er in from this Sectioncase .. . 8 period Indeed 2 period .. our However Definition a detailed study5 . 1 of 2 algebraic of conformality structure invariant heavily under dependsparabolic rota on thehyphen 2 . \quad The lengths from centres ( 5 . 5 ) are conformal for any combination of values of $ \sigma , t ionsunderlying .. open square linear bracket structure 4 8 comma in a ..: we 4 7 measure closing square a distance bracket ..between removes points .. obscurityy and fromy + thisty0 ..and case period .. Indeed .. our \breve{\sigma} $ and $ \ mathringR {\sigma} { . }$ Definitionintuitively .. 5 period expect 1 2 ..of that conformality it i s always small for small t. As explained in [ 4 8 , 3heavily . \quad dependsThe on lengths the underlying from linear foci structure\quad ( in 5 R . to 6 the ) power\quad ofare a : we conformal measure a distance for $ between\ mathring points{\sigma}\not= 0 $ \quady andand y plus any ty combination to the power of of prime values and intuitively of $ \ expectsigma that$ it i s always small for small t period .. As explained in .. open square bracket 4 8 comma \ [ and \breve{\sigma} { . }\ ]

\noindent Proof . \ h f i l l This i s another straightforward calculation in G i NaC [ 4 6 , \S $ 3 . 5 . 2 ] . \ blacksquare $

\noindent The conformal property of the distance ( 5 . 2 ) −− ( 5 . 3 ) \quad from Proposition 5 . 1 3 . 1 i s well − known , \quad o f course , \quad see \quad [ 1 6 , \quad 7 2 ] . \quad However \quad the \quad same \quad property \quad o f non − symmetric \quad l e n g t h s \quad from \quad Proposi − t i o n s \quad 5.13.2and5.1 \quad 3 . 3 could b e hardly expected . \quad The smaller group $ SL { 2 } ( R ) ($ incomparison

\noindent t o all linear − fractional transforms of $ R ˆ{ 2 } ) $ generates bigger number of conformal metrics , cf . Re − mark 3 . 5 .

The \quad exception of the case \quad $ \ mathring {\sigma} = 0 $ from the conformality in \quad 5 . 1 3 . 3 \quad l o o k s \quad disappointing on the first glance , \quad especially in the light of the parabolic Cayley transform considered lat er in Sec tion \quad 8 . 2 . \quad However a detailed study of algebraic structure invariant under parabolic rota − t i o n s \quad [ 4 8 , \quad 4 7 ] \quad removes \quad obscurity from this \quad case . \quad Indeed \quad our Definition \quad 5 . 1 2 \quad of conformality heavily depends on the underlying linear structure in $ R ˆ{ a } : $ we measure a distance between points $y$ and $y + tyˆ{\prime }$ and intuitively expect that it i s always small for small $ t . $ \quad As explained in \quad [ 4 8 , 28 .... V period V period Kisil \noindenthline28 28 \ h f i l l V . V . K i s i l V . V . Kisil S 3 period 3 closing square bracket the standard linear structure i s incompatible with the parabolic rotations and thus should \ [ b\ er replacedu l e {3em by}{ a0.4 more pt }\ relevant] one period .. More precisely comma instead of limits y to the power of prime right arrow y along the straight lines t owards y we need t o consider limits along vertical lines comma see Fig period .. 1 8 and open square bracket 4 8 comma Fig period 2 and §3 . 3 ] the standard linear structure i s incompatible with the parabolic rotations and thus \noindentshould\S 3 . 3 ] the standard linear structure i s incompatible with the parabolic rotations and thus should Remark 3 period 1 7 closing square bracket period 0 Propositionb e replaced 5 period by 14 a period more .. relevant Let the focal one .length More is given precisely by the identity , instead open of parenthesis limits y 5→ periody along 6 closing the parenthesis .. with sigma =\noindent ring-sigmastraight =b 0 e lines comma replaced t owards .. e period byy awe g more period need relevant t o consider one limits . \quad alongMore vertical precisely lines , see , instead Fig . 1 of 8 and limits $ y ˆ{\prime } \rightarrowl sub[ 4 f sub 8 , Fig sigma-brevey . 2 $ and along to Remark the power the 3 straight of . 2 1 open 7 ] . parenthesis y comma y to the power of prime closing parenthesis = minus breve-sigma p to thelines powerProposition t of owards 2 minus 2 $ 5 vp y . comma $ 14 we . where need pLet= topen o the consider parenthesisfocal length limits u to is the given along power byof vertical the prime identity minus lines u( closing 5 , . see 6 parenthesis ) Figwith . \ toquad the power18and of 2 divided [ 48 , Fig . 2and byRemark 2 openσ = parenthesis 3˚σ = . 0 1, 7e ]v . to g . . the power of prime minus v closing parenthesis sub period Then it is conformal in the s ense that for any constant y = ue sub 0 plus( veu0 − subu) 12 .. and y to the power of prime = u to the power of prime e\noindent sub 0 plus vProposition to the power of 5 prime . 14l e2 sub( .y,\ y1quad0) .. = with−σp˘Let2 − the2vp, focal where length p = is given by the identity ( 5 . 6 ) \quad with fσ˘ 2(v0 − v) $ \asigma fixed u to= the power\ mathring of prime{\ wesigma have} : = 0 , $ \quad e . g . . Equation:Then it open is conformalparenthesis 5 in period the s 10 ense closing that parenthesis for any .. constant limint primey = v rightue + arrowve infinityand ly sub0 = u f0 sube + sigma-brevev0e open parenthesis g \ [ l ˆ{ 2 } { f {\breve{\sigma}}} ( y , y ˆ{\prime0 } 1) = − \breve0 {\1sigma} p ˆ{ 2 } − times ywith comma a g fixed times yu to0 we the have power : of prime closing parenthesis divided by l sub f sub breve-sigma open parenthesis y comma y to the power of2 prime vp closing , parenthesis where = p 1 divided = \ f by r a copen{ ( parenthesis u ˆ{\ cuprime plus d} closing − parenthesisu ) ˆ{ to2 the}}{ power2 of ( 2 sub v comma ˆ{\prime where g} = −Row 1v a be ) } { . }\ ] sub 0 Row 2 minus ce sub 0 d . period l (g · y, g · y0) 1  a be  We also revise the paraboliclim casefσ˘ of conformality= in Section 6where period 2 g with= a related definition0 . based (5.10) \noindent Then it is0 conformal→∞ in0 the s ense2 that for any constant−ce d $ y = ue { 0 } + ve { 1 }$ on infinitesimal cycles periodv lfσ˘ (y, y ) (cu + d) , 0 \quadRemarkand 5 period $ y ˆ 1{\ 5 periodprime .. The} = expressions u ˆ{\ ofprime lengths} opene parenthesis{ 0 } + 5 period v ˆ{\ 5 closingprime parenthesis} e { comma1 }$ open\quad parenthesiswith 5 period 6a closing f i x e dparenthesisWe $ also u ˆ revise{\ areprime generally the parabolic}$ non we hyphen have case symmetric : of conformality and this i sin Section 6 . 2 with a related definition a pricebased one onshould infinitesimal pay for its non cycles hyphen . triviality period All symmetric distances lead t o nine two hyphen dimensional \ beginCayleyRemark{ a endash l i g n ∗} Klein 5 . 1 geometries 5 . commaThe expressions.. see .. open square of lengths bracket ( 57 2 . comma 5 ) , ( .. 5 Appendix . 6 ) are B closing generally square non bracket - comma .. open square bracket\limsymmetric 2{\ .. 6prime comma and ..{ 2 thisv .. 5}\ i closing s arightarrow price square one bracket should\ periodinfty pay .. for}\ In itsthef r a non parabolic c { -l triviality{ casef a{\ symmetric . Allbreve symmetric{\sigma}}} distances( g \cdot y , g distance\leadcdotof t o ay ninevector ˆ{\ two openprime - parenthesisdimensional} ) }{ u commal Cayley{ vf closing –{\ Kleinbreve parenthesis geometries{\sigma i s always}}} , see a( function [ y 7 2 of , , u alone Appendix y ˆ comma{\prime B cf ] period ,} ) ..} Remark= \ 5f rperiod a c { 1 2 }{ ( 3cu period +[ 2 .. For d 6 such , ) a2ˆ{ distance2 5}} ] .{ , In} thewhere parabolic g case = a\ symmetricl e f t (\ begin { array }{ cc } a & be { 0 }\\ − ce { 0 } & d \aend parabolicdistance{ array unit}\ ofright circle a vector consists). (u,\ tag from v) i∗{ stwo$ always (vertical 5 a lines function . .. open 10 of parenthesis )u alone $} , see cf dotted . Remark vertical lines 5 . in 2 the 3 . second For \endrows{sucha lon i g Figsn a∗} distance period 8 and a parabolic 1 .. 1 closing unit parenthesis circle consists comma from which two i s not vertical aesthetically lines attractive ( see dotted period ..vertical On the other hand the parabolic quotedblleftlines in theunit second cycles quotedblright defined by lengths open parenthesis 5 period 5 closing parenthesis and open parenthesis 5 period 6 Weclosing alsorows parenthesis revise on Figs are the . parabolas 8 parabolic and 1 comma 1 case ) which , which of makes conformality i s the not parabolic aesthetically Cayleyin Section attractive 6 . . 2 with On athe related other hand definition based ontransform infinitesimalthe parabolic open parenthesis “ cycles unit see cycles . Section ” defined .. 8 period by 2 lengths closing parenthesis ( 5 . 5 ) and very (natural 5 . 6 period ) are parabolas , which Wemakes can also the consider parabolic a distance Cayley between points in the upper half hyphen plane which is preserved by \noindentM o-dieresistransformRemark bius ( transformationssee 5 Section . 1 5 . \ comma 8quad . 2 )The see very open expressions natural square . bracket of 3 lengths2 closing square ( 5 . bracket 5 ) period, ( 5 . 6 ) are generally non − symmetric and this i s aLemma priceWe 5 one period can should also 16 period consider pay .... for Let a distance the its line non e between lement− triviality be points bar dy bar in . the to All the upper symmetric power half of 2 - = plane distances du to which the power islead preserved of 2 t minus o nine sigma two dv− to thedimensional power of 2Cayley .... andby the−− M ....Kleino¨ bius quotedblleft transformations geometries length of , a\ curve ,quad see quotedblright [see 3 2 ]\ .quad is[ given 7 2 , \quad Appendix B ] , \quad [ 2 \quad 6 , \quad 2 \quad 5 ] . \quad In the parabolic case a symmetric 2 2 2 by theLemma corresponding 5 . 16 line . integralLet the comma line e .. lement cf period be .. open| dy | square= du bracket− σdv 6and comma the S 1“ 5 length period of 2 closing a curve square ” bracket comma \noindentintegralis given subdistance Capital Gamma of a bar vector dy bar $(divided uby v sub , period v open) $ parenthesis i s alwaysa 5 period function 1 1 closing of parenthesis $u$ alone , cf . \quad Remark 5 . 2 3 . \quad For such a distance aThen parabolicby the the length corresponding unit of the circle curve line is consists preserved integral under , from the cf two . M o-dieresis vertical[ 6 , §1 bius 5 lines . transformations 2 ] ,\quad ( period see dotted vertical lines in the second R |dy| Proof period .. The proof i s based on the following three observations : Γ v . (5.11) \noindent1 periodThen .. therows The length line on element Figs of the . .. curve 8bar and dy is bar 1 preserved\ toquad the power1 under ) of, which2 the = du M to i the so¨ bius not power transformations aesthetically of 2 minus sigma . attractive dv to the power . of\quad 2 .. atOn .. the the point other .. e hand the parabolic sub‘‘ 1 unit ..Proof i s invariant cycles . underThe ’’ defined actionproof of i s theby based lengths on the ( following 5 . 5 ) three and ( observations 5 . 6 ) are : parabolas , which makes the parabolic Cayley 2 2 2 resp ective1 . fix hyphen The line group element of this point| dy open| = parenthesisdu − σdv see Lemmaat 2 the period point 8 closinge1 parenthesisi s invariant period under \noindent2 periodaction ..transform Theof the fraction (bar see dy bar Section divided\ byquad v is invariant8 . 2 ) under very action natural of the .ax plus b hyphen group period 3 periodresp ective.. M dieresis-o fix - group bius action of this of point SL sub ( 2 see open Lemma parenthesis 2 . 8 R ) closing . parenthesis in each EPH case i s generated by ax plus b group Weand canthe corre also hyphen consider a distance|dy between| points in the upper half − plane which is preserved by 2 . The fraction v is invariant under action of the ax + b− group . Msp $ onding\ddot fix{o hyphen} $ biussubgroup transformations comma see Lemma , 2 seeperiod [ 1 3 0 2 period ] . blacksquare 3 . M o¨ bius action of SL2(R) in each EPH case i s generated by ax + b group and the It iscorre known - open sp onding square bracketfix - subgroup 6 comma , S see 1 5 Lemma period 2 closing2.10. square bracket in the elliptic case that the curve b etween two points with the\noindent shortestIt isLemma known 5 [ . 6 16 , §1 . 5\ .h f2 i l] l inLet the the elliptic line case e thatlement the be curve $ b\mid etweendy two points\mid ˆ with{ 2 } the= du ˆ{ 2 } − \sigma dv ˆ{ 2 }$ \ h f i l l and the \ h f i l l ‘‘ length of a curve ’’ is given lengthshortest open parenthesis length ( 5 5 .period 1 1 ..) i 1 s closing an arc parenthesis of the circle i s an orthogonal arc of the circle t o the orthogonal real line t o. the M realo¨ linebius period .. M o-dieresis bius transformationstransformations map map \noindentsuchsuch arcs toarcsby arcs the to with arcs corresponding the with same the property same line comma property integral therefore , therefore the , length\quad the ofc such length f . arc\quad ofcalculated such[ 6 arc in , open calculated\S 1 parenthesis 5 . 2in ] ( 5 5 period , . 1 1 closing parenthesis i s 1 1 ) i s invariant under the M o¨ bius transformations . \ hspaceinvariant∗{\Analogously underf i l l } the$ M\ inint o-dieresis the{\ hyperbolic biusGamma transformations}\ casef r a c the{\ periodmid longestdy curve\mid b etween}{ v } two{ . points} ( isan 5 arc . of 1 1 ) $ Analogouslyhyperbola in the orthogonal hyperbolic caset o the longest real line curve . b However etween two in points the parabolic is an arc of case hyperbola there i s no curve \noindentorthogonaldeliveringThen t o the the the real shortest length line period of However the curve in the isparabolic preserved case there under i s no the curve M delivering $ \ddot the{o shortest} $ bius transformations . lengthlength open ( parenthesis 5 . 1 1 ) 5 , period the infimum 1 1 closing is parenthesisu − u0, see comma ( 5 . the 3 ) infimum and Fig is u . minus 1 0 . u However to the power we of can prime still comma see open parenthesis 5\noindent perioddefine 3 closingProof an invariantparenthesis . \quad anddistanceThe Fig proofperiod in the 1 i 0 parabolic speriod based However caseon the inwe the can following stillfollowing define three an way invariant : observations : distanceLemma in the 5 parabolic . 1 7 ( case [ 3 in 2 the ] ) following . Let way two : points w1 and w2 in the upper half - plane are \ hspaceLemmalinked∗{\ 5 periodf by i l lan}1 arc 7 . open\ ofquad parenthesisThe line open element square bracket\quad 3 2 closing$ \mid squaredy bracket\mid closingˆ{ parenthesis2 } = period du ˆ{ ....2 Let} two − points \sigma w sub 1dv ˆ{ 2 }$ \quad at \quad the point \quad $ e { 1 }$ \quad i s invariant under action of the .... anda w parabola sub 2 in the with upper zero halfσ˘ hyphen− radius plane . are Then linked the by an length arc of ( 5 . 1 1 ) along the arc is invariant a parabolaunder Mwith zeroo¨ bius sigma-breve transformations hyphen radius . period .. Then the length open parenthesis 5 period 1 1 closing parenthesis .. along the arc is\noindent invariant underresp M ective o-dieresis fix bius− group of this point ( see Lemma 2 . 8 ) . transformations period \ centerline {2 . \quad The fraction $\ f r a c {\mid dy \mid }{ v }$ is invariant under action of the $ ax + b − $ group . }

3 . \quad M $ \ddot{o} $ bius action of $ SL { 2 } ( R ) $ in eachEPHcase i s generated by $ ax + b $ group and the corre − sp onding fix − subgroup , seeLemma $2 . 1 0 . \ blacksquare $

It is known [ 6 , \S 1 5 . 2 ] in the elliptic case that the curve b etween two points with the shortest length ( 5 . 1 \quad 1 ) i s an arc of the circle orthogonal t o the real line . \quad M $ \ddot{o} $ bius transformations map

\noindent such arcs to arcs with the same property , therefore the length of such arc calculated in ( 5 . 1 1 ) i s invariant under the M $ \ddot{o} $ bius transformations .

Analogously in the hyperbolic case the longest curve b etween two points is an arc of hyperbola orthogonal t o the real line . However in the parabolic case there i s no curve delivering the shortest

\noindent length ( 5 . 1 1 ) , the infimum is $ u − u ˆ{\prime } , $ see ( 5 . 3 ) and Fig . 1 0 . However we can still define an invariant distance in the parabolic case in the following way :

\noindent Lemma5 . 17( [ 32 ] ) . \ h f i l l Let two points $ w { 1 }$ \ h f i l l and $ w { 2 }$ in the upper half − plane are linked by an arc of

\noindent a parabola with zero $ \breve{\sigma} − $ r a d i u s . \quad Then the length ( 5 . 1 1 ) \quad along the arc is invariant under M $ \ddot{o} $ bius transformations . Erlangen Program at Large hyphen 1 : .... Geometry of Invariants .... 29 \noindenthlineErlangenErlangen Program Program at Large at - Large 1 :− 1 : \ h f i l l GeometryGeometry of of Invariants Invariants \ h f i l l 29 29 Figure 1 1 period .. Focal orthogonality in all nine combinations period To highlight both similarities and distinctions \ [ with\ r u lthe e {3em ordinary}{0.4 orthogonality pt }\ ] we use the same notations as that in Fig period .. 8 period .. The cycles C-tilde sub breve-sigma to the power of sigma-breve .. from PropositionFigure 1 4 1 period . Focal 2 4 are orthogonality drawn by dashed in all lines nine periodcombinations . To highlight both similarities and distinctions ˜σ˘ \noindent5 periodwith the 3Figure .. ordinary Perpendicularity 1 orthogonality 1 . \quad and orthogonality weFocal use the orthogonality same notations as in that all in Fig nine . combinations 8 . The cycles C .σ˘ Tofrom highlight both similarities and distinctions withIn aProposition Euclidean the ordinary space 4 . 2 4 the orthogonalityare shortest drawnby distance dashed we from lines use a . point the t same o a line notations i s provided as by thethat corresp in Fig onding . \quad 8 . \quad The c y c l e s $ \ tilde {C} ˆ{\breve{\sigma}} {\breve{\sigma}}$ \quadp erpendicular5from . 3 Perpendicularity period .... Since we have and already orthogonality defined various distances and lengths we may use them forIn a definition a Euclidean of corresponding space the notions shortest of p distance erpendicularity from a period point t o a line i s provided by the corresp \noindentif functionondingProposition from by minus-A-minus 4 . 2 sub4 are arrowright-B drawn by leftthreetimes dashed lines minus-C-minus . arrowright-D sub period to right arrow from a vec minus- C-minusp Derpendicular to Definition .5 denoted Since to the we power have of already period 18 defined period various Let l to the distances power of and b l to lengths the power we of may e open use parenthesis to the power of right\noindent arrowthem from5 . minus-A-minus 3 \quad Perpendicularity to a sub B to the power and oforthogonality length plus epsilon or right arrow minus-C-minus sub D closing parenthesis distance sub offor a variable a definition period We of corresponding say epsilon has that notions a sub of local p erpendicularity a vec sub extremum . to the power of minus-A-minus sub arrowright-B i s l sub at byminus−A−minusarrowright−B minus−C−minusarrowright−D. minus−A−minus length \noindent In a Euclidean space theh shortest distance fromb e →a a point t o a line i s provided by the corresp onding hypheniffunction bottom epsilona~minus =−C 0−minusD period This. t18 o. i sub s l ( B +εor→minus − C − minusD)distanceofavariable. →Definition5 denoted Letl Remark 5 period 19 period minus−A−minusarrowright−B \noindentWe saypεhas erpendicularthatalocal a ~extremum . \ h f i l l Since wei have s l@ − already ⊥ε=0. This definedtois various distances and lengths we may use them 1 periodRemark Obviously 5 . 19 minus-C-minus . D-arrowright leftthreetimes minus-A-minus sub B-arrowright sub closing parenthesis the l hyphen p erpendicularity similarly sub to f hyphen orthogonality sub comma to the power of i s not a sub see symmetric Subsection notion 4 period 3 1. Obviously )thel − period\noindent open parenthesisfor a definition i period e period of corresponding right arrow minus-A-minusminus− notionsC−minusD sub− ofarrowright B p leftthreetimes erpendicularityhminus−A right−minus arrowB− .arrowright minus-C-minus D does not imply perpendicularitysimilarlyisnot a symmetric notion ( i . e . → minus − A − minus → Equation: l non sub hyphentof 0−orthogonality to the power, see of hyphenSubsection p erpendicularity4.3. r period is Howevernot sub necessarilyBh imply to the power of obviously\noindentminus l r right−$C if− arrowminusD function minus-A-minusdoes not ˆ{ implyby sub B minus hyphen−A p−minus linear in{ erpendicularityarrowright − leftthreetimesB }\ leftthreetimes right arrow minus-C-minus minus−C− Dminus period right arrowright −D { . }} {\rightarrow ˆ{ a arrow\vec{} from periodminus e− periodC−minus minus-A-minus D } { De sub fi B to nition right arrow 5 from}{ idenoted s not generally}ˆ{ linear. 18 in minus-A-minus . } Let sub l B} commaˆ{ b } to minus-C-minusl ˆ{ e } ( ˆ{\rightarrow ˆ{ minus−A−minus } { a }}ˆ{ length } { B } +D-arrowright\ varepsilon sub commaor i period{\ erightarrow period minus-A-minus}{ minus sub−C B-arrowright−minus } { leftthreetimesD) } minus-C-minusd i s t a n c e { D-arrowrightof a variable implies minus-A-minus} . $ We $ say {\ varepsilon has } that { a } { l2 o. c a l }$ a $ \vec{} { extremum }ˆ{ minus−A−minus { arrowright −B }}$ −perpendicularity obviously .e.minus−A−minusB sub arrowright-B sub i to the power of leftthreetimes r minus-C-minus D-arrowright right arrowhminus from− leftthreetimesC−minusDdoes minus-C-minus D does to lnon −0r. isHowevernotnecessarilyimplylr→minus−A−minusB − plinearinerpendicularity →minus−C−minusD. → isnotgenerallylinearinminus−A−minusB, i s $ l { @ } −{ \bot } {\ varepsilon = 0 . This } t o →{ foranyreali } { s }$ h → for any real .. 2 period hrminus−C−minusD−arrowright minus−C−minusD−arrowright,i.e.minus−A−minusB−arrowrighthminus−C−minusD−arrowrightimpliesminus−A−minusarrowright−B i \noindent Remark 5 . 19 .

\ hspace ∗{\ f i l l } $1 . Obviously { minus−C−minus D−arrowright \ leftthreetimes minus−A−minus { B−arrowright }} { ) } the l − p erpendicularity { s i m i l a r l y } { to f − orthogonality }ˆ{ i s not } { , } a { see } symmetric { Subsection } notion { 4 . 3 . } ( $ i . e $ . \rightarrow{ minus−A−minus } { B } \ leftthreetimes \rightarrow{ minus−C−minus } D $ does not imply

\ begin { a l i g n ∗} 2 . \ tag ∗{$ l { non }ˆ{ − p erpendicularity } { − 0 } r . i s { However }{ not }ˆ{ obviously } { necessarily imply } l { r \rightarrow{ minus−A−minus } { B }} − p l i n e a r in { erpendicularity {\ leftthreetimes \rightarrow{ minus−C−minus } D. }}\rightarrow ˆ{ . e . minus−A−minus { B }} {\rightarrow ˆ{ i s not generally linear in minus−A−minus { B, }} { minus−C−minus D−arrowright { , } i . e . minus−A−minus { B−arrowright }\ leftthreetimes minus−C−minus D−arrowright implies minus−A−minus { arrowright −B }} { i }ˆ{\ leftthreetimes } r minus−C−minus D−arrowright }\rightarrow ˆ{\ leftthreetimes minus−C−minus D does } { f o r any r e a l }$} \end{ a l i g n ∗} 30 .... V period V period Kisil \noindenthline30 30 \ h f i l l V . V . K i s i l V . V . Kisil There is the following obvious connection b etween perpendicularity and orthogonality period \ [ Lemma\ r u l e { 53em period}{0.4 20 ptperiod}\ ] .... Let minus-A-minus sub B-arrowright .... be l sub c sub sigma-breve hyphen perpendicular open parenthesis l sub f breve-sigma hyphen perpendicular closing parenthesis to a vector minus-C-minus D-arrowright sub period .... Then the flat right arrowThere from is the through following B period obvious The vec connection minus-C-minus b etween D is tangent perpendicularity to C sub sigma and to the orthogonality power of s to . cycle open parenthesis straight \ centerlineLemma{There 5 . 20 is . the followingLet minus obvious− A − minus connection b etweenbe perpendicularityl − perpendicular and(l − orthogonality . } underline closing parenthesis AB comma is open parenthesis s hyphenB−arrowright closing parenthesiscσ˘ orthogonal to sub at thefσ˘ B period .... cycle .... C sub sigma toperpendicular the power of s ) .... to with a vector .... centreminus .... open− C parenthesis− minusD − focusarrowright closing. parenthesis .... at A ....Then passing the flat throughB. T he~minus−C−minusDistangenttoCs \noindent Lemma 5 . 20 . \ h f i l l Letσ $ minus−A−minus {s B−arrowright }$ \ h f i l l be $ l { c {\breve{\sigma}}} Proof→cycle period(straight .. This follows) AB, from is the(s− relation)orthogonal of centreto@the ofB. opencycle parenthesisCσ with s hyphen centre closing parenthesis( focus ghost) at cycleA to centre open parenthesis −focus$ closingpassing perpendicular parenthesis of $ open ( parenthesis l { f s hyphen\breve closing{\sigma parenthesis} } − ortho$ perpendicular hyphen ) to a vector $ minus−C−minus D−arrowrightgonalProof cycle stated . { in.This Propositions}$ follows\ h f i l l ..fromThen 4 period the the relation9 .. flat and .. of 4 periodcentre 2 of 4 corresp ( s - ) ondingly ghost cycle period to blacksquare centre ( focus ) Consequentlyof ( s - ) orthothe p erpendicularity - gonal cycle of stated vectors in minus-A-minus Propositions sub B-arrowright 4 . 9 and and minus-C-minus 4 . 2 4 corresp D-arrowright ondingly is reduced t o the orthog- onality\noindent. of $ \rightarrow ˆ{ through B . The \vec{} minus−C−minus D is tangent to Cˆ{ s } {\sigma }} { c y c l e ( s t r a i g h t \ ) AB , i s ( s − ) orthogonal } to { @ } the { B. }$ \ h f i l l c y c l e \ h f i l l the correspondingConsequently flat cycles the p only erpendicularity in the cases comma of vectors when orthogonalityminus − A − itselfminus iB s− reducedarrowright t oand the localminus − C − $ Cnotion ˆminusD{ s at} the{\− pointarrowrightsigma of cycles}$ is intersections\ reducedh f i l l with t openo the\ parenthesish orthogonality f i l l c e n see t r e Remark\ ofh f ithe l .. l 4 corresponding( period f o c u s8 closing ) \ hf flat parenthesis i l l cyclesat $ period only A $ in\ h f i l l passing Obviouslythe cases comma , when l hyphen orthogonality perpendicularity itself turns i s reduced .. t o .. be t o .. the the .. local usual .. orthogonality in the .. elliptic .. case comma .. cf period \noindentLemmanotion 5 periodProof at the 22 . point period\quad of openThis cycles parenthesis follows intersections e closing from ( parenthesis the see Remark relation b elow of4 period . centre8 ) .. . For of two ( other s − cases) ghost the description cycle to i s given centre as follows ( focus : ) of ( s − ) ortho − gonalLemma cycleObviously 5 period stated 2 1 period, in l− ..perpendicularity Propositions Let A = open parenthesis\quad turns4 u . tcomma 9 o\quad be v closingand the parenthesis\quad usual4 .. . and orthogonality2 4 B corresp = open parenthesis inondingly the u to $ the . power\ blacksquare of prime $ commaelliptic v to the power case of , prime cf . closing parenthesis period .. Then Consequently1 periodLemma d hyphen 5 the . 22 perpendicular p . ( erpendicularity e ) b elow open . parenthesis For of two vectors in other the s ense cases $ of minus open the− description parenthesisA−minus 5{ i period sB given−arrowright 2 closing as follows parenthesis}$ : and closing $ parenthesisminus−C−minus to right Darrow−arrowright minus-A-minusLemma 5$ . subis 2 1 reducedB . in theLet elliptic t oA theor= hyperbolic(u, orthogonality v) and casesB is= a ( mulu of0, v0 hyphen). Then the corresponding flat cycles only in the cases , when orthogonality itself i s reduced t o the local tiple of1 the. d vector− perpendicular ( in the s ense of ( 5 . 2 ) ) to → minus − A − minusB in the elliptic openor parenthesishyperbolic sigmacases open is parenthesis a mul - tiplev minus of thev to vector the power of prime closing parenthesis to the power of 3 minus open parenthesis u\noindent minus u tonotion the power at of the prime point closing of parenthesis cycles to intersections the power of 2 open ( see parenthesis Remark v\quad plus v4 to .the 8 power ) . of prime open parenthesis 1 minus 2 sigma sigma-breve closing parenthesis closing parenthesis comma breve-sigma open parenthesis u minus u to the power of prime closing 0 3 0 2 0 0 3 0 0 0 0 parenthesis\ hspace ∗{\ tof the i( lσ l( power}vObviously− v ) of− 3( minusu − u $ open) ,(v + parenthesisv l (1 −−2σ$σ˘)) u, perpendicularity minusσ˘(u − u) to− the(u power− u )(v of turns− primev )(−\2 closingquadv + (vt parenthesis+ ov )˘\σσquad)), openbe parenthesis\quad the v\ minusquad vusual to the \quad orthogonality in the \quad e l l i p t i c \quad case , \quad c f .

power of prime closing parenthesis open parenthesis minus 2 v to the power of prime plus0 open parenthesis0 v plus v to the power of prime closing parenthesis\noindent sigma-breveLemma5which sigma . 22 forclosing . (σ e parenthesisσ˘ )= b 1 reduces elow closing . to\quad the parenthesis expectedFor two comma value other(v cases− v , σ( theu − u description)). i s given as follows : which for2 sigma. d− sigma-breveperpendicular = 1 reduces( in the to sense the expected of ( 5 value . 3 ) open ) to parenthesisminus − vA minus− minus v toB the−arrowright power ofin prime comma sigma open parenthesis\noindentthe uparabolicLemma minus u 5 tocase . the 2 power is 1 . (0\ of,quad t) prime, t ∈ RLet$A closingwhich parenthesis = closing( u parenthesis , v period )$ \quad and $ B = ( u ˆ{\prime } ,2period v ˆ{\ dprime hyphenco} perpendicular incides) . with $ open\ thequad parenthesisGalileanThen in orthogonality the sense of opendefined parenthesis in [ 7 5 2 period , §3 ] 3. closing parenthesis closing parenthesis .. to 3. l − perpendicular ( in the sense of ( 5 . 5 ) ) to minus − A − minus is a multiple of (σv0 − ˚σv, u − u0). cσ˘ minus-A-minus sub B-arrowright in the parabolic case is open parenthesis 0 commaarrowright t closing−B parenthesis comma t in R which $co 1 incides . with d the− Galilean$ perpendicular orthogonality defined ( in the in open sense square6 of bracket ( 5 7 . 2 2 comma ) ) S to 3 closing $\rightarrow square bracket{ minus period−A−minus } { B }$ 4. lfσ˘ − perpendicular ( in the s ense of ( )) to minus − A − minusarrowright−B is a in3 the period elliptic l sub c sub or sigma-breve hyperbolic hyphen cases perpendicular is a mul open− 5 parenthesis. in the sense of open parenthesis 5 period 5 closing parenthesis 0 0 closingtiplemultiple parenthesis of the of vector.. to(σv minus-A-minus+ p, u − u ), subwhere arrowright-Bp is a multiple of open parenthesis sigma v to the power of prime minus sigma-ring v comma u minusis u defined to the power either of primeby ( closing5 . 7 ) parenthesisor by period( 5 . 8 ) for corresponding values of ˚σ. \ begin4 periodProof{ a l il g sub n . ∗} f sub sigma-breve hyphenThe perpendicular perpendiculars open parenthesis are calculated in the by s enseG i of NaC open[ paranthesisRow 4 6 , § 3.5.3]. 1 6 Row 2 5 period . closing parenthesis( \sigmaIt .. is to worth minus-A-minus( t v o have− sub anv arrowright-B idea ˆ{\ aboutprime is} a different multiple) ˆ{ 3 of types} open − parenthesis of( perpendicularity u sigma− vu to ˆ the{\ in powerprime the oft erms} prime) ofplus ˆ{ 2 p comma} ( u v minus + u to v ˆ{\prime } the( power 1the of− prime2 closing\sigma parenthesis\breve comma{\sigma .. where} p )), \breve{\sigma} ( u − u ˆ{\prime } ) ˆ{ 3 } − is defined(standard u either− Euclidean byu open ˆ{\ parenthesisprime geometry} 5 period.) Here ( 7 closing vare someparenthesis− v examples ˆ{\ .. orprime by . open} parenthesis)( 5− period2 8 closing v ˆ{\ parenthesisprime } for+ corresponding ( v +values v ofLemma ˆ ring-sigma{\prime 5 . sub} 22 period .) \Letbreveminus{\sigma− A}\− minussigmaB−arrowright)),= ue0 + ve1 and minus − C − minusD − 0 0 \endProof{arrowrighta l periodi g n ∗} ....= Theu e0 perpendiculars+ v e1, then : are calculated by G i NaC open square bracket 4 6 comma S 3 period 5 period 3 closing square bracket period blacksquare ( e ) . In the e lliptic case the d− perpendicularity for σ˘ = −1 means that \ centerlineIt is→ worthminus t{ owhich− haveA − anminus f o idea rB about$and\sigma ..→ differentminus\breve types− C −{\ ..minusDsigma of perpendicularity}form= a right 1 in $ the reduces t erms of to the the expected value $ ( v − 0 0 v ˆstandard{\prime Euclidean} , geometry\sigma periodangle( .. Here u , are− or some analyticallyu ˆexamples{\prime perioduu} +)vv = ) 0. . $ } Lemma 5 period 22 period .. Let minus-A-minus( p ). Intheparabolic sub B-arrowrightGminus = ue−C− subminusD 0 plus−arrowrightand ve sub 1 ..casethe and minus-C-minusvertical the D-arrowright = u to the \ hspace ∗{\ f i l l } $ 2 . d − $ perpendicular ( in the sense of ( 5 . 3 ) ) \quad to $ minus−A−minus { B−arrowright }$ power oflfσ˘ prime− perpendicularity e sub 0 plus vdirectionoranalytically to the power of prime: for e subσ˘ = 1 comma 1 means then : that → minus − A − minusB bisect intheparaboliccaseisopenthe parenthesis angle e closing parenthesis $( period 0.. , In tthe e ) lliptic , case t the d\ in hyphenR perpendicularity $ which for sigma-breve = minus 1 means that right arrow minus-A-minus sub B and right arrow minus-C-minus D form a right \ centerlineangle comma{ co .. or incides analytically with uu to the the Galilean power of prime orthogonality plus vv to the power defined of prime in [= 70 period 2 , \S 3 ] . } 0 0 0 0 p open parenthesis p closing parenthesisu periodu − v Inp = theu u parabolic− v ( u between2 + v2 − subv) = minus-C-minus 0, D-arrowright and(5. case12) the sub vertical the l sub f\ subcenterline sigma-breve{ $ hyphen 3 . perpendicularity l { c {\ directionbreve or{\sigma analytically} }} : for− $ breve-sigma perpendicular = 1 .. means ( in that the right sense arrow of minus-A-minus ( 5 . 5 ) sub ) \ Bquad .. to bisect$ minus the− ..A− angleminus { arrowrightwhere−B }$p isis thea multiple focal length of( $ 5 ( . 7 )\sigma v ˆ{\prime } − \ mathring {\sigma} vEquation: , u open− parenthesisu ˆ{\prime 5 period} 1 2) closing . $ parenthesis} .. u to the power of prime u minus v to the power( of h prime p = u to the power the hyperbolic the −perpendicularity of prime). u Inminus minus v to− theA − powerminus ofB− primearrowright parenleftbigminus radicalbig-line−C−minusD of− uarrowrightare to the powercasebisected of 2 plus v todby the power of 2 minus v parenrightbig = 0 \ hspace ∗{\ f i l l } $ 4 . l { f and{\breve{\sigma} }} − $ perpendicular ( inlines the s ense of $(\ begin { array }{ c} 6 \\ comma = 1 meansthat G parallel to foru=σ˘±v, − oranalytically 0utheu−anglesv0v=0. 5where . \end p is{ thearray focal}) length ) $ open\quad parenthesisto $5 period minus 7−A closing−minus parenthesis{ arrowright −B }$ is a multiple of $ ( \sigma v ˆ{\prime } + pRemark , u 5 .− 23 .u ˆ{\prime } If) one attempts, $ \quad to devisewhere a $ parabolic p $ length as a limit or an openintermediate parenthesis h case closing parenthesis period In minus-A-minus sub B-arrowright to the power of the sub and to the power of hyperbolic sub minus-C-minus D-arrowright2 are2 case bis ected to the power2 of the2 d by to the power of hyphen sub lines to the power of perpendicularity \ centerlinefor the{ ellipticis definedle = u either+ v and by hyperbolic ( 5 . 7 )lp \=quadu − vorlengths by ( 5 then . 8 the ) for only corresponding possible guess i values s of $ \ mathring {\sigma} { . }$ parallel0 to for2 u = sigma-breve plusminux v sub comma to the power of = minus or sub analytically to the power of 1 to the power of means lp = u (5.3), which i s t oo trivial for an interesting geometry . that} prime u the u minus angles v to the power of prime v = 0 sub period to the power of between Similarly the only orthogonality conditions linking the elliptic u1u2 +v1v2 = 0 Remark 5 period 23 period .... If one attempts to devise a parabolic length as a limit or an intermediate case \noindentandProof the hyperbolic . \ h f i l l uThe1u2 − perpendicularsv1v2 = 0 cases seems are to calculated b e u1u2 = 0(bysee G [i 7 NaC 2 , § [3 4] and 6 , Lemma\S $ 35 . 5 . 3 ] for. the 2 elliptic 1 . 2 ) l , sub which e = u i to s againthe power t oo of trivial 2 plus v . to the This power support of 2 and our hyperbolic Remark l sub 1 p . = 1 u . to 2 .the power of 2 minus v to the power of 2. lengths\ blacksquare then the only $ possible guess i s l sub p to the power of prime = u to the power of 2 open parenthesis 5 period 3 closing parenthesis comma which i s t oo trivial for an interesting\ hspace ∗{\ geometryf i l l } It period is worth t o have an idea about \quad different types \quad of perpendicularity in the t erms of the Similarly the .. only .. orthogonality .. conditions .. linking the .. elliptic u sub 1 u sub 2 plus v sub 1 v sub 2 = 0 .. and .. the \noindenthyperbolicstandard u sub 1 u sub Euclidean 2 minus v sub geometry 1 v sub 2 . =\quad 0 casesHere seems are to b e some u sub examples 1 u sub 2 = . 0 open parenthesis see open square bracket 7 2 comma S 3 closing square bracket and Lemma 5 period 2 1 period 2 closing parenthesis comma which \noindenti s again tLemma oo trivial 5 period . 22 .. . This\quad supportLet our $ Remark minus− ..A− 1minus period 1{ periodB−arrowright 2 period } = ue { 0 } + ve { 1 }$ \quad and $ minus−C−minus D−arrowright = u ˆ{\prime } e { 0 } + v ˆ{\prime } e { 1 } , $ then :

\ hspace ∗{\ f i l l }( e ) . \quad In the e lliptic case the $ d − $ perpendicularity for $ \breve{\sigma} = − 1 $ means that $\rightarrow{ minus−A−minus } { B }$ and $\rightarrow{ minus−C−minus } D$ form a right

\ centerline { angle , \quad or analytically $ uu ˆ{\prime } + vv ˆ{\prime } = 0 . $ }

\ hspace ∗{\ f i l l }(p $) . In the parabolic {\ between } { minus−C−minus D−arrowright and } case { the } { v e r t i c a l }$ the $ l { f {\breve{\sigma} }} −{ perpendicularity } { direction or analytically : }$ f o r $ \breve{\sigma} = 1 $ \quad means that $\rightarrow{ minus−A−minus } { B }$ \quad b i s e c t the \quad angle

\ begin { a l i g n ∗} u ˆ{\prime } u − v ˆ{\prime } p = u ˆ{\prime } u − v ˆ{\prime } ( \ sqrt { u ˆ{ 2 } + v ˆ{ 2 }} − v ) = 0 , \ tag ∗{$ ( 5 . 1 2 ) $} \end{ a l i g n ∗}

\ centerline {where $ p $ is the focal length ( 5 . 7 ) }

\ hspace ∗{\ f i l l }( h $ ) . In { minus−A−minus } { B−arrowright }ˆ{ the } { and }ˆ{ h y p e r b o l i c } { minus−C−minus D−arrowright are } case { b i s } ected ˆ{ the } d{ by }ˆ{ − }ˆ{ perpendicularity } { l i n e s }$ parallel to $ f o r { u = }\breve{\sigma}{\pm } v ˆ{ = } { , } −{ or }ˆ{ 1 } { analytically }ˆ{ means that }\prime { u } the { u − } a n g l e s { v ˆ{\prime } v = 0 }ˆ{\ between } { . }$

\noindent Remark 5 . 23 . \ h f i l l If one attempts to devise a parabolic length as a limit or an intermediate case

\noindent for the elliptic $ l { e } = u ˆ{ 2 } + v ˆ{ 2 }$ and hyperbolic $ l { p } = u ˆ{ 2 } − v ˆ{ 2 }$ lengths then the only possible guess i s

\noindent $ l ˆ{\prime } { p } = u ˆ{ 2 } ( 5 . 3 ) , $ which i s t oo trivial for an interesting geometry .

Similarly the \quad only \quad orthogonality \quad c o n d i t i o n s \quad l i n k i n g the \quad e l l i p t i c $ u { 1 } u { 2 } + v { 1 } v { 2 } = 0 $ \quad and \quad the hyperbolic $ u { 1 } u { 2 } − v { 1 } v { 2 } = 0$ cases seems tobe $u { 1 } u { 2 } = 0 ($ see[72, \S 3 ] andLemma5 . 2 1 . 2 ) , which i s again t oo trivial . \quad This support our Remark \quad 1 . 1 . 2 . Erlangen Program at Large hyphen 1 : .... Geometry of Invariants .... 3 1 \noindenthlineErlangenErlangen Program Program at Large at - Large 1 :− 1 : \ h f i l l GeometryGeometry of of Invariants Invariants \ h f i l l 3 1 3 1 6 .. Invariants .. of infinitesimal .. scale \ [ Although\ r u l e {3em parabolic}{0.4 zeropt }\ hyphen] radius cycles defined in Definition 3 period 1 2 do not satisfy our expectations for .. quotedblleft zero hyphen radius quotedblright .. but they are oft en t echnically suitable for the same purposes as elliptic and hyperbolic6 Invariants ones period .. Yet we of may infinitesimal want t o find something scale which fit s b etter for our intuition on \noindentquotedblleftAlthough6 zero\quad parabolic siz edI n quotedblright v a zero r i a n - t sradius\ ....quad obj cycles ectof period infinitesimal defined .... inHere Definition we present\quad 3 an .s c 1approach a 2 l e do not based satisfy on non our hyphen expecta- Archimedean open parenthesis non hyphentions standard for “ closing zero - parenthesis radius ” but they are oft en t echnically suitable for the same purposes \noindentanalysisas elliptic openAlthough square and hyperbolicbracket parabolic 1 8 comma ones zero . ..− 6 9Yetradius closing we may square cycles want bracket defined t o period find something in Definition which 3 fit . s 1 b 2etter do for not satisfy our expectations f o6 r periodour\quad intuition 1 ..‘ Infinitesimal ‘ zero on − r radius a d i u s cycles ’ ’ \quad but they are oft en t echnically suitable for the same purposes as elliptic and hyperbolicLet“ epsilon zero siz be ones a ed positive ” . \ objquad infinitesimal ectYet . Here we number may we wantpresentcomma t i periodano find approach e period something based 0 less n on which epsilon non less- fit Archimedean 1 sfor b any etter n in (N for non open our - square intuition bracket 1 8 on comma 6 9 closingstandard square ) bracket period \noindentDefinitionanalysis 6‘‘ period [ zero 1 8 , 1 siz period 6 9 ed ] .... . ’’ A\ cycleh f i l lC subobj sigma-breve e c t . \ h fto i l the l Here power we of s present such that an det approach C sub breve-sigma based on to the non power− Archimedean of s i s an ( non − standard ) infinitesimal6 . 1 number Infinitesimal i s called infinitesimal radius cycles \noindentLet ε analysisbe a positive [ 1 infinitesimal 8 , \quad number6 9 ] . , i . e .0 < nε < 1 for any n ∈ N[1 8, 69]. radius cycle period s s LemmaDefinition 6 period 2 6 period . 1 . .. Let sigma-breveA cycle andCσ˘ such sigma-ring that detbe twoCσ˘ metrici s an signs infinitesimal and let a point number open i parenthesis s called u sub 0 comma v sub 0 closing\noindentinfinitesimal parenthesis6 . 1in\ Rquad to theInfinitesimal power of p with v radiussub 0 greater cycles 0 period .. Consider radius cycle . a cycle C sub sigma-breve to the power of s defined by p \noindentLemmaLet 6 .$ 2\ varepsilon . Let $σ˘ and be a˚σ positivebe two metric infinitesimal signs and numberlet a point , i .(u0 e, v0) $∈ .R with 0 < n \ varepsilon Equation: open parenthesis 6 period 1 closings parenthesis .. C sub sigma-breve to the power of s = parenleftbig 1 comma u sub 0 comma n u to0 f. o the rConsider any power $ of n 2 plus a cycle\ in 2 nv subNCσ˘ 0defined minus [ 1sigma-ring by 8 ,n to the 6 power 9 of ] 2 parenrightbig . $ comma where \noindentEquation:Definition open parenthesis 6 6. period 1 . \ 2h closing f i l l A parenthesis c y c l e $ .. C n = ˆ{ braceex-braceex-braceleftmid-braceex-braceex-braceleftbts } {\breve{\sigma}}$ such that det $Cˆ v sub{ 0 epsilons } {\breve{\sigma}}$ Cs = (1, u , n, 02 + 2nv − ˚σn2), (6.1) subi s 2 an divided infinitesimal by 2 v sub 0 minus number radicalbig-line i s calledσ˘ of 0 v to infinitesimal0 the poweru of 20 minus open parenthesis sigma-ring minus sigma-breve closing parenthesis epsilonwhere to the power of 2 divided by comma sigma-ring minus sigma-breve to the power of comma if to the power of if ring-sigma to the power of\noindent sigma-ring equal-negationslashradius cycle . = sigma-breve sub period to the power of breve-sigma sub comma Then , \noindent Lemma 6 . 2 . \quad Let $ \breve{\sigma} $ and $ \ mathringp 2 {\sigma2 } $ be two metric signs and let a point 1 period .. The point open parenthesis u sub 0 comma v sub 0 closing parenthesisv0ε2 ..− is ring-sigma0 − (˚σ − σ hyphen˘)ε focus of the cycle period $ (n u= braceex{ 0 } − braceex, v −{braceleftmid0 } ) −\ inbraceexR− ˆbraceex{ p }$− braceleftbt with $ v { 0 } v > 0 . $if if\quad˚σ˚σ 6=Considerσ˘σ˘, 2 period .. The square of sigma-breve hyphen radius is exactly minus epsilon to2 thev power, of˚σ 2− commaσ˘ i period e period= open. parenthesis 6 a c y c l e $ C ˆ{ s } {\breve{\sigma}}$ d e f i n e d by 0 period 1 closing parenthesis .. defines an infinitesimal radius cycle period (6.2) 3 period .. The focal length of the cycle is an infinitesimal number of order epsilon to the power of 2 period \ beginProofThen{ perioda l i g n ∗} .. The cycle open paranthesisRow 1 6 Row 2 period 1 . has the squared sigma-breve hyphen radius equal to minus epsilon to C ˆ{ s } {\breve{\sigma}} = ( 1 , u { 0 } , n , 0 { u }ˆ{ 2 } + 2 nv { 0 } − the power of 2 if n is a root1 to . the equation The point : (u0, v0) is ˚σ− focus of the cycle . 2 .\ mathringopen The parenthesis square{\sigma of ring-sigma} σ˘−nradius ˆ{ minus2 } is breve-sigma exactly), \ tag− closingε2,∗{i$ . parenthesis e ( . ( 6 6 . n .1 to ) the 1defines power ) $ of} an 2 minus infinitesimal 2 v sub 0 n radius plus epsilon cycle to . the power of 2 = 0 period\end{3a l . i g n ∗} The focal length of the cycle is an infinitesimal number of order ε2. Proof . The Moreover only6 the root from open parenthesis 6 period 2 closing parenthesis of the quadratic case gives an infinitesimal focal length period cycle ( ) has the squared σ˘− radius equal to −ε2 if n is a root to the equation : ..\noindent This where.1 also i s supported by calculations done in G i N aC comma see open square bracket 4 6 comma S 3 period 6 period 1 closing square bracket (˚σ − σ˘)n2 − 2v n + ε2 = 0. period\ begin blacksquare{ a l i g n ∗} 0 nThe =Moreover graph braceex of cycle only− openbraceex the parenthesis root−braceleftmid from 6 ( period 6 . 2 1 )− closing ofbraceex the parenthesis quadratic−braceex in case− thebraceleftbt parabolicgives an space infinitesimal\ drawnf r a c { atv focalthe{ scale0 length}{\ of realvarepsilon numbers looks} { like2 }}{ 2 v { 0 }}\ f r a c { − \ sqrt { 0 { v }ˆ{ 2 } − ( \ mathring {\sigma} − \breve{\sigma} ) \ varepsilon ˆ{ 2 }}}{ , a vertical. This ray started also i at s supported its focus comma by calculations see Fig period done .. 12 in openG i Nparenthesis aC , see a [ closing 4 6 , § parenthesis3.6.1].  comma due t o the following lemma period\ mathringThe{\sigma graph} of cycle − (\breve 6 . 1 ){\ insigma the parabolic}}ˆ{ , } spacei f drawn ˆ{ i f at}\ themathring scale of real{\sigma numbers} ˆ{\ looksmathring {\sigma}}\ne { = } \breveLemmalike{\sigma 6 a period vertical} 3ˆ{\ .. ray openbreve started parenthesis{\sigma at its open} focus{ square, ,}} see bracket{ Fig. }\ . 4tag 6 comma 1∗{ 2$ ( (a .. ) S , .. 6 due 3 period t. o the 62 period following ) $ 1} closing lemma square . bracket closing parenthesis period\end{Lemma ..a l Infinitesimal i g n ∗} 6 . 3 cycle ..( open[ 4 6 parenthesis , § 3 6 . period 6 . 1 1 ] closing ) . parenthesisInfinitesimal .. consists cycle of points( 6 comma . 1 ) ..consists which are infinitesi hyphen mallyof c points los e open , parenthesis which are in infinitesi the sense -of length mally from c los focus e ( openin the parenthesis sense of 5 lengthperiod 6 from closing focus parenthesis( 5 . 6 closing parenthesis .. to its \noindent Then focus F)) = opento parenthesis its focus uF sub= (0u comma0, v0): v sub 0 closing parenthesis : Equation: open parenthesis 6 period 3 closing parenthesis .. parenleftbigg u sub 0 plus epsilon u comma v sub 0 plus v sub 0 u to the power \ centerline {1 . \quad Thepoint $ ( u { 0 } , v { 0 } ) $ \quad i s $ \ mathring {\sigma} − $ of 2 plus parenleftbig open parenthesis sigma-breve minus sigma-ring closing2 parenthesis u to the power of 2 minus ring-sigma parenrightbig focus of the cycle . } 2 2 ε 3 epsilon to the power of 2 divided by( 4u0 v+ subεu, 0 v plus0 + v O0u parenleftbig+ ((˘σ − ˚σ)u epsilon− ˚σ) to the+ O power(ε )). of 3 parenrightbig parenrightbigg(6.3) period Note that points below of F open parenthesis in the ordinary scale closing4v0 parenthesis are not infinitesimally close to F in the \ centerlinesense of length{2 open . \quad parenthesisThe square 5 period of6 closing $ \breve parenthesis{\sigma comma} but − are$ in radius the sense is of exactly distance open $ − parenthesis \ varepsilon 5 period 3ˆ{ closing2 } ,$ i.e.(6.1)Note that points\quad belowdefines of F ( in anthe infinitesimal ordinary scale ) radius are not cycleinfinitesimally . } close to F in parenthesisthe period .. Fig period 1 2 open parenthesis a closing parenthesis shows elliptic comma hyperbolic concentricsense of and length parabolic ( 5 confocal . 6 ) , cyclesbut are of decreasing in the sense radii of which distance shrink ( t 5 o .the 3 corresp ) . ondingFig . 1 2 ( a ) shows \noindentinfinitesimal3 radius . \quad cyclesThe period focal length of the cycle is an infinitesimal number of order $ \ varepsilon ˆ{ 2 } . $ elliptic , hyperbolic concentric and parabolic confocal cycles of decreasing radii which shrink It it so easy the t ocorresp see that onding infinitesimal infinitesimal radius cycles radius has properties cycles . similar t o zero hyphen radius ones comma cf period ProofLemma . 3\ periodquad 1The .. 4 c period y c l e $(\ begin { array }{ c} 6 \\ . 1 \end{ array })$ has the squared $ \breve{\sigma} − $ radiusIt i s easy equal t o to see $that− infinitesimal \ varepsilon radiusˆ{ 2 cycles}$ has if properties $n$ is similar a root t oto zero the - radius equation ones : , cf . Lemma 3 . 1 4 . \ [( \ mathring {\sigma} − \breve{\sigma} ) n ˆ{ 2 } − 2 v { 0 } n + \ varepsilon ˆ{ 2 } = 0 . \ ]

\noindent Moreover only the root from ( 6 . 2 ) of the quadratic case gives an infinitesimal focal length . \quad This also i s supported by calculations done in G i NaC , see [ 4 6 , \S $ 3 . 6 . 1 ] . \ blacksquare $

The graph of cycle ( 6 . 1 ) in the parabolic space drawn at the scale of real numbers looks like a vertical ray started at its focus , see Fig . \quad 1 2 ( a ) , due t o the following lemma .

\noindent Lemma 6 . 3 \quad ( [ 4 6 , \quad \S \quad 3 . 6 . 1 ] ) . \quad Infinitesimal cycle \quad ( 6 . 1 ) \quad consists of points , \quad which are infinitesi − mally c los e ( in the sense of length from focus ( 5 . 6 ) ) \quad toitsfocus $F = ( u { 0 } , v { 0 } ) : $

\ begin { a l i g n ∗} ( u { 0 } + \ varepsilon u , v { 0 } + v { 0 } u ˆ{ 2 } + ( ( \breve{\sigma} − \ mathring {\sigma} ) u ˆ{ 2 } − \ mathring {\sigma} ) \ f r a c {\ varepsilon ˆ{ 2 }}{ 4 v { 0 }} + O( \ varepsilon ˆ{ 3 } )). \ tag ∗{$ ( 6 . 3 ) $} \end{ a l i g n ∗}

\ hspace ∗{\ f i l l }Note that points below of $ F ( $ in the ordinary scale ) are not infinitesimally close to $ F $ in the

\noindent sense of length ( 5 . 6 ) , but are in the sense of distance ( 5 . 3 ) . \quad Fig . 1 2 ( a ) shows elliptic , hyperbolic concentric and parabolic confocal cycles of decreasing radii which shrink t o the corresp onding infinitesimal radius cycles .

It i s easy t o see that infinitesimal radius cycles has properties similar t o zero − radius ones , cf . Lemma 3 . 1 \quad 4 . 32 .... V period V period Kisil \noindenthline32 32 \ h f i l l V . V . K i s i l V . V . Kisil Figure .. 12 period .. open parenthesis a closing parenthesis .. Zero hyphen radius cycles in elliptic .. open parenthesis black point closing parenthesis\ [ \ r u l e { ..3em and}{ hyperbolic0.4 pt }\ open] parenthesis the red light cone closing parenthesis period .. In hyphen finitesimal radius parabolic cycle is the blue vertical ray starting at the fo cus period .. open parenthesis b closing parenthesis Elliptic hyphen parabolicFigure hyphen 12 . ( a ) Zero - radius cycles in elliptic ( black point ) and hyperbolic ( the red light \noindenthyperboliccone ) .Figure phase In - transition finitesimal\quad between12 radius . \quad parabolic fixed points( a cycle ) of\ theisquad the subgroup blueZero vertical K− periodradius ray starting cycles at the in fo cuselliptic . ( b )\ Ellipticquad ( black point ) \quad and hyperbolic ( the red light cone ) . \quad In − finitesimalLemma- parabolic 6 period radius - hyperbolic 4 period parabolic .... phase The transition image cycle of SL between is sub the 2 fixed open blue points parenthesis vertical of the R subgroup closing ray starting parenthesisK. at hyphen the action fo cus on an . infinitesimal\quad ( b radius ) Elliptic cycle − p a r a b o l i c − ....hyperbolic openLemma parenthesis phase 6 . 6 4 period transition. 1The closing image parenthesis between of SL fixed ....2(R by)− conjugaaction points hyphen on of an the infinitesimal subgroup radius $ K cycle . $ ( 6 . 1 ) tionby open conjuga parenthesis - 3 period 4 closing parenthesis .. is an infinitesimal radius cycle of the same order period \noindentImagetion of an(Lemma 3 infinitesimal . 4 )6 .is 4 cycle an . \ infinitesimal underh f i l l cycleThe conjugation image radius of cycle is an $ of SLinfinitesimal the{ same2 } cycle order(R) of . the same − $ action on an infinitesimal radius cycle \ h f i l l ( 6 . 1 ) \ h f i l l by conjuga − or lesserImage order ofperiod an infinitesimal cycle under cycle conjugation is an infinitesimal cycle of the same \noindentProofor period lessert i o .... norder These( 3 . . are 4 calculations ) \quad is done an in infinitesimal G i NaC comma see radius open square cycle bracket of the 4 6 same comma order S 3 period . 6 period 2 closing square bracketProof period blacksquare. These are calculations done in G i NaC , see [ 4 6 , § 3.6.2].  ImageThe consideration ofThe an infinitesimal consideration of infinitesimal of cycle infinitesimal numbers under in the numbers cycle elliptic conjugation and in hyperbolic the elliptic case is and an should hyperbolic infinitesimal not case should cycle not of the same orbring lesserbring any advantages any order advantages . since the since open the parenthesis ( leading leading ) quadratic closing parenthesis t erms quadraticin these t cases erms in are these non cases - zero are non . hyphen zero period .... HoweverHowever \noindentnonnon hyphen - ArchimedeanProof Archimedean . \ h f inumbers l l These in in the are the parabolic parabolic calculations case case provide provide done a more in a moreintuitive G i NaC intuitive and , efficient see and [ efficientpresen 4 6 , hyphen\ presenS $ 3 - . 6 . 2 ] . tation\tationblacksquare period . .. For For $ example example zero zero hyphen - radius radius cycles cycles are are not helpful helpful for for the the parabolic parabolic Cayley Cayley transform transform open parenthesis see Subsection( see .. 8 period 2 closing parenthesis but infinitesimal cycles are their successful replacements period \ hspaceTheSubsection second∗{\ f parti l l } ofThe the 8 . consideration following 2 ) but infinitesimalresult i s of a useful infinitesimal cycles substitution are their for numbers successful Lemma 4 in period replacements the 5 elliptic period . and hyperbolic case should not Lemma 6The period second 5 period part .... of Let the C sub following sigma-breve result to i the s a power useful of substitutions .... be the infinitesimal for Lemma cycle 4 . .... 5 open . parenthesis 6 period 1 closing s ˘s parenthesis\noindentLemma ....bring and 6 breve-C . any 5 . advantages subLet sigma-breveCσ˘ be since the to the infinitesimal the power ( of leading s = cycle open ) parenthesis quadratic( 6 . 1 ) k comma tand erms lC commaσ˘ in= ( thesek, n l, comman, m cases) be m aclosing are non parenthesis− zero .... . be\ h a f i l l However genericgeneric cycle period cycle . \noindentThenThen non − Archimedean numbers in the parabolic case provide a more intuitive and efficient presen − s ˘s t a1 t periodi o n . ..\quad The orthogonalityFor1 . example The condition orthogonalityzero − .. openradius parenthesis condition cycles 4 are period(4. not2) 2 closingC helpfulσ˘ ⊥Cσ˘ parenthesisand for the the C f -sub parabolic orthogonality sigma-breve Cayley to the transform power of s bottom ( see ˘s s breve-C(4 sub.12) sigma-breveCσ˘ a Cσ˘ toboth the power of s .. and the f hyphen orthogonality .. open parenthesis 4 period 1 2 closing parenthesis breve-C sub sigma-breve\noindent toSubsection the power of s\quad turnstileright8 . 2 C ) sub but breve-sigmaare infinitesimal given to by the power cycles of s .. are both their successful replacements . are given by \ centerline {The second part of the following2 result i s a useful substitution for Lemma 4 . 5 . } ku sub 0 to the power of 2 minus 2 lu sub 0 plusku0 m− =2lu O0 open+ m = parenthesisO(ε). epsilon closing parenthesis period In other words the cycle C-breve sub˘s breve-sigma to the power of s has root u sub 0 in the parabolic space period \noindentIn otherLemma words 6 the . 5 cycle . \ h fC iσ˘ l lhasLet root $u C0 in ˆ{ thes } parabolic{\breve space{\sigma . 2}} .$ The\ h f if l - l orthogonalitybe the infinitesimal cycle \ h f i l l ( 6 . 1 ) \ h f i l l and 2 period(4.12) ..Cs Thea C˘ fs hyphenis given orthogonality by open parenthesis 4 period 1 2 closing parenthesis C sub sigma-breve to the power of s turnstileright breve-C$ \breve sub{C sigma-breve}σ˘ ˆ{ σ˘s } to{\ thebreve power{\ ofsigma s is given}} by=(k,l,n,m)$ \ h f i l l be a generic cycle . Equation: open parenthesis 6 period 4 closing parenthesis .. ku sub 0 to the power of 2 minus 2 lu sub 0 minus 2 nv sub 0 plus m = O open parenthesis\noindent epsilonThen closing parenthesis period 2 ku0 − 2lu0 − 2nv0 + m = O(ε). (6.4) In other words the cycle C-breve sub breve-sigma to the power of s passes focus open parenthesis u sub 0 comma v sub 0 closing parenthesis ..\ hspace of the infinitesimal∗{\ f i l l }1 cycle . \quad in the parabolicThe orthogonality˘s condition \quad $ ( 4 . 2 ) C ˆ{ s } {\breve{\sigma}} In other words the cycle Cσ˘ passes focus (u0, v0) of the infinitesimal cycle in the \botspace period\breve{C} ˆ{ s } {\breve{\sigma}}$ \quad and the f − orthogonality \quad $ ( 4 . 1 2 ) \parabolicbreve{C} ˆ{ s } {\breve{\sigma}}\dashv C ˆ{ s } {\breve{\sigma}}$ \quad both Proof period .... These are G i NaC calculations openspace square . bracket 4 6 comma S 3 period 6 period 3 closing square bracket period blacksquare 3.6.3]. \ centerlineProof{ . are given by } These are G i NaC calculations [ 4 6 , §  It is interestingIt is interesting to note that to thenote exotic that f the hyphen exotic orthogonality f - orthogonality b ecame warranted b ecame replacement warranted of replacement the of usual one for the infinitesimal cycles period \ [ kuthe ˆ{ 2 } { 0 } − 2 lu { 0 } + m = O ( \ varepsilon ). \ ] usual one for the infinitesimal cycles .

\noindent In other words the cycle $ \breve{C} ˆ{ s } {\breve{\sigma}}$ has root $ u { 0 }$ in the parabolic space . 2 . \quad The f − orthogonality $( 4 . 1 2 ) Cˆ{ s } {\breve{\sigma}}\dashv \breve{C} ˆ{ s } {\breve{\sigma}}$ i s given by

\ begin { a l i g n ∗} ku ˆ{ 2 } { 0 } − 2 lu { 0 } − 2 nv { 0 } + m = O ( \ varepsilon ). \ tag ∗{$ ( 6 . 4 ) $} \end{ a l i g n ∗}

\ hspace ∗{\ f i l l } In other words the cycle $ \breve{C} ˆ{ s } {\breve{\sigma}}$ passes focus $ ( u { 0 } , v { 0 } ) $ \quad of the infinitesimal cycle in the parabolic

\ centerline { space . }

\noindent Proof . \ h f i l l These are G i NaC calculations [ 4 6 , \S $ 3 . 6 . 3 ] . \ blacksquare $

\ hspace ∗{\ f i l l } It is interesting to note that the exotic f − orthogonality b ecame warranted replacement of the

\noindent usual one for the infinitesimal cycles . Erlangen Program at Large - 1 : Geometry of Invariants 33

6 . 2 Infinitesimal conformality An intuitive idea of conformal maps , which i s oft enly provided in the complex analysis t extbooks for illustration purposes , is “ they send small circles into small circles with resp ective centres ” . Using infinitesimal cycles one can turn it into a precise definition . Definition 6 . 6 . A map of a region of Ra t o another region i s l− infinitesimally conformal for a length l( in the sense of Definition 5 . 5 ) if for any l− infinitesimal cycle : 1 . It s image i s an l− infinitesimal cycle of the same order . 2 . The image of it s centre / focus is displaced from the centre / focus of it s image by an infinitesimal number of a greater order than it s radius . Remark 6 . 7 . Note that in comparison with Definition 5 . 1 2 we now work “ in the opposite direction ” : former we had the fixed group of motions and looked for corresponding conformal lengths / distances , now we take a distance / length ( encoded in the infinitesimally equidistant cycle ) and check which motions respect it . Natural conformalities for lengths from centre in the elliptic and parabolic cases are already well studied . Thus we are mostly interested here in conformality in the parabolic case , where lengths from focus are b etter suited . The image of an infinitesimal cycle ( 6 . 1 ) under SL2(R)− action is a cycle , moreover it s i s again an infinitesimal cycle of the same order by Lemma 6 . 4 . This provides the first condition of Definition 6 . 6 . The second part is delivered by the following

statement :

˘s s Proposition 6 . 8 . Let Cσ˘ be the image under g ∈ SL2(R) of an infinitesimal cycle Cσ˘ from ( 6 . 1 ) . ˘s 2 Then ˚σ− focus of Cσ˘ is displaced from g(u0, v0) by infinitesimals of order ε ( while both cycles have σ˘− radius of order ε). Consequently SL2(R)− action is infinitesimally conformal in the s ense of Definition 6 . 6 with respect to the length from focus ( Definition 5 . 5 ) for all combinations of σ, σ˘ and ˚σ. Proof . These are G i NaC calculations [ 4 6 , § 3.6.2].  Infinitesimal conformality seems intuitively t o b e close to Definition 5 . 1 2 . Thus it i s desir - able t o give a reason for the absence of exclusion clauses in Proposition 6 . 8 in comparison t o Proposition 5 . 1 3 . 3 . As shows calculations [ 4 6 , §3 . 5 . 2 ] the limit ( 5 . 9 ) at point y0 = u0e0 + v0e1 do exist but depends from the direction y = ue0 + ve1 :

d(g · y0, g · (y0 + ty)) 1 lim = (6.5) t→0 2 2 2 d(y0, y0 + ty) (d + cu0) + σc 0v − 2Kcv0(d + cu0) ,  a b  where K = u and g = . However if we consider points ( 6 . 3 ) of the infinitesimal v c d cycle then 0+ve1. Italso thevalue infinitesimal from K v u εu coincides of(the limit (6.5) at the scale y= 0 =line−two ue=ε . Thus upto aninfinitesimal number )with thevalue v0u i s independentin(5.10). Remark 6 . 9 . There is another connection between parabolic function theory and non - standard analysis . As was mentioned in Section 2 , the Clifford algebra C − lscript(p) corresponds to the set of dual numbers u + εv with ε2 = 0[72, Supplement C ] . On the other hand we may consider the set of numbers u + εv within the non - standard analysis , with ε b eing an infinitesimal . In this case ε2 is a higher order infinitesimal than ε and effectively can b e treated as 0 at infinitesimal scale of ε, i . e . we again get the dual numbers condition ε2 = 0. This explains why many results of differential calculus can b e naturally deduced within dual numbers framework [ 1 1 ] . Erlangen Program at Large hyphen 1 : .... Geometry of Invariants .... 33 \noindenthline InfinitesimalErlangen cycles Program are at also Large a convenient− 1 : \ th ool f i l l forGeometry calculations of ofInvariants invariant measures\ h f i l l 33 , Jaco 6 period- bians 2 .. ,Infinitesimal etc . conformality \ [ An\ r uintuitive l e {3em idea}{0.4 of ptconformal}\ ] maps comma which i s oft enly provided in the complex analysis t extbooks for illustration purposes comma is .. quotedblleft they send small circles into small circles with resp ective centres quotedblright period Using infinitesimal cycles one can turn it into a precise definition period \noindentDefinition6 6 period . 2 \ 6quad periodInfinitesimal .... A map of a region conformality of R to the power of a t o another region i s l hyphen infinitesimally conformal for a length l open parenthesis in the sense of Definition 5 period 5 closing parenthesis if for any l hyphen infinitesimal cycle : \noindent1 period ..An It s intuitive image i s an lidea hyphen of infinitesimal conformal cycle maps of the , which same order i s period oft enly provided in the complex analysis t extbooks for2 period illustration .. The .. image purposes .. of it s .. , centre is \quad slash focus‘‘ they .. is .. send displaced small .. from circles .. the .. into centre small slash focus circles .. of it with s .. image resp .. by ective .. an centres ’’ . Usinginfinitesimal infinitesimal number of a cyclesgreater order one than can it turn s radius it period into a precise definition . Remark .. 6 period 7 period .. Note that in comparison with .. Definition .. 5 period 1 2 we now work .. quotedblleft in the opposite \noindentdirection quotedblrightDefinition : ..6 former . 6 . we\ h had f i l the l Amap fixed group of a of regionmotions and of looked $Rˆ for{ correspondinga }$ t o anotherconformal region i s $ l − $ infinitesimallylengths slash distances conformal comma .. for now we take a distance slash length .. open parenthesis encoded in the infinitesimally equidistant cycle closing parenthesis and check which motions respect it period \noindentNatural conformalitiesa length for $ lengths l ( from $ in centre the in sense the elliptic of and Definition parabolic cases 5 . are 5 )already if for any $ l − $ infinitesimal cycle : well studied period .. Thus we are mostly interested here in conformality in the parabolic case comma where \ centerlinelengths from{1 focus . \ arequad b etterIt suited s image period i .. s The an image $ l of an− infinitesimal$ infinitesimal cycle open cycle parenthesis ofthe 6 period same 1 closing order parenthesis . } under SL sub 2 open parenthesis R closing parenthesis hyphen 2 .action\quad is aThe cycle\ commaquad image moreover\quad it s i so again f i t an s infinitesimal\quad centre cycle of/ the focus same\quad order byi s Lemma\quad 6 periodd i s p l a 4 c period e d \quad from \quad the \quad centre / focus \quad o f i t s \quad image \quad by \quad an infinitesimalThis provides the number first condition of a of greater Definition order 6 period than 6 period it s.. The radius second . part is delivered by the following statement : \noindentPropositionRemark 6 period\quad 8 period6 Let . 7 C-breve . \quad subNote breve-sigma that to in the comparison power of s be with the image\quad underD e g f iin n iSL t i o sub n \ 2quad open parenthesis5 . 1 2 we R closing now work \quad ‘‘ in the opposite parenthesisdirection of an ’’ infinitesimal : \quad former cycle C sub we sigma-breve had the fixed to the power group of sof from motions open parenthesis and looked 6 period for 1 closing corresponding parenthesis periodconformal lengthsThen ring-sigma / distances hyphen focus , \quad of breve-Cnow wesub sigma-brevetake a distance to the power / length of s .... is\quad displaced( encoded from g open in parenthesis the infinitesimally u sub 0 comma v equidistantsub 0 closing parenthesis .... by infinitesimals of order epsilon to the power of 2 open parenthesis while both cycles \noindenthave sigma-brevecycle hyphen ) and radius check of whichorder epsilon motions closing respect parenthesis it period . Consequently SL sub 2 open parenthesis R closing parenthesis hyphen action is infinitesimally conformal in the s ense of Definition 6 period 6Natural .. with conformalities for lengths from centre in the elliptic and parabolic cases are already wellrespect studied to the length . \quad from focusThus open we parenthesisare mostly Definition interested 5 period here 5 closing in parenthesis conformality for all combinationsin the parabolic of sigma comma case, sigma-breve where andlengths sigma-ring from sub focus period are b etter suited . \quad The image of an infinitesimal cycle ( 6 . 1 ) under $ SL { 2 } (R)Proof period− ....$ These are G i NaC calculations open square bracket 4 .... 6 comma S 3 period 6 period 2 closing square bracket period blacksquare \noindentInfinitesimalaction conformality is a seems cycle intuitively , moreover t o b e it close s to i Definition s again 5 an period infinitesimal 1 2 period .. Thus cycle it i s of desir the hyphen same order by Lemma 6 . 4 . Thisable t provides o give a reason the for first the absence condition of exclusion of Definition clauses in Proposition 6 . 6 . ..\ 6quad periodThe 8 in secondcomparison part t o is delivered by the following Proposition 5 period 1 3 period 3 period .. As shows calculations open square bracket 4 6 comma S 3 period 5 period 2 closing square bracket the\ begin limit{ opena l i g nparenthesis∗} 5 period 9 closing parenthesis at point y 0 = u sub 0 e sub 0 plus v sub 0 e sub 1 statementdo exist but depends : from the direction y = ue sub 0 plus ve sub 1 : \endEquation:{ a l i g n ∗} open parenthesis 6 period 5 closing parenthesis .. limint t right arrow 0 d open parenthesis g times y 0 comma g times open parenthesis y 0 plus ty closing parenthesis closing parenthesis divided by d open parenthesis y 0 comma y 0 plus ty closing parenthesis = 1 divided\noindent by openProposition parenthesis d 6 plus . cu8 sub. Let 0 closing $ \breve parenthesis{C} toˆ{ thes power} {\ ofbreve 2 plus{\ sigmasigma c}} to the$ power be the of 2 image 0 v to the under power $of 2g minus\ in 2 KcvSL sub{ 2 0 open} ( parenthesis R ) d $ plus of cu an sub infinitesimal 0 closing parenthesis cycle sub comma $Cˆ{ s } {\breve{\sigma}}$ from(6.1) . where K = u divided by v and g = Row 1 a b Row 2 c d . period .... However if we consider points open parenthesis 6 period 3 closing parenthesis\noindent ofThen the infinitesimal $ \ mathring cycle {\sigma} − $ f o c u s o f $ \breve{C} ˆ{ s } {\breve{\sigma}}$ \ h f i l l is displaced from $ gfrom ( to the u power{ 0 of} then, Ky v ={ v sub0 } 0 u) from $ 0\ plush f i l ve l subby 1 infinitesimals period It also to = of line-two order sub$ v sub\ varepsilon 0 u to the powerˆ{ 2 of} epsilon( $ u ue while = both cycles epsilon underbar sub period Thus coincides to the power of the value of open parenthesis the up to limit an infinitesimal open parenthesis 6 period\noindent 5 closinghave parenthesis $ \breve at the{\ subsigma number} sub − $ closing radius parenthesis of order to the power $ \ varepsilon of infinitesimal sub) with . scale$ sub the value i s independent in open parenthesis 5 period 1 0 closing parenthesis period ConsequentlyRemark 6 period $ 9 SL period{ ..2 There} (R) is another connection− $ action between parabolic is infinitesimally function theory andconformal non hyphen in standard the s ense of Definition 6 . 6 \quad with respectanalysis period to the .. As length was mentioned from focus in Section ( Definition 2 comma theClifford 5 . 5 algebra ) for C-lscript all combinations open parenthesis of p closing $ \sigma parenthesis, ..\ correspondsbreve{\sigma} $ toand the set $ \ ofmathring {\sigma} { . }$ dual numbers u plus epsilon v with epsilon to the power of 2 = 0 open square bracket 7 2 comma Supplement C closing square bracket period\noindent On theProof other hand . \ h we f i may l l These consider are the G i NaC calculations [ 4 \ h f i l l 6 , \S $ 3 . 6 . 2 ] . \ blacksquareset of numbers $ u plus epsilon v within the non hyphen standard analysis comma with epsilon b eing an infinitesimal period .. In this case epsilon to the power of 2 is a higher order infinitesimal than epsilon and effectively can b e treated as 0 at infinitesimal \ hspacescale of∗{\ epsilonf i l l comma} Infinitesimal i period e period conformality we again get the seems dual intuitivelynumbers condition t o epsilon b e to close the power to Definition of 2 = 0 period 5 .. . This 1 2 explains . \quad whyThus it i s desir − many results \noindentof differentialable calculus t o can give b e a naturally reason deduced for the within absence dual numbers of exclusion framework clausesopen square in bracket Proposition 1 .. 1 closing\quad square6 bracket . 8 in period comparison t o PropositionInfinitesimal cycles 5 . are 1 3 also . a3 convenient . \quad tAs ool shows for calculations calculations of invariant [ 4measures 6 , \S comma3 . Jaco5 . hyphen 2 ] the limit ( 5 . 9 ) at point $ ybians 0 comma = etc u period{ 0 } e { 0 } + v { 0 } e { 1 }$ do exist but depends from the direction $ y = ue { 0 } + ve { 1 } : $

\ begin { a l i g n ∗} \lim { t \rightarrow 0 }\ f r a c { d ( g \cdot y 0 , g \cdot ( y 0 + ty ) ) }{ d(y0,y0+ty) } = \ f r a c { 1 }{ ( d + cu { 0 } ) ˆ{ 2 } + \sigma c ˆ{ 2 } 0 { v }ˆ{ 2 } − 2 Kcv { 0 } ( d + cu { 0 } ) } { , }\ tag ∗{$ ( 6 . 5 ) $} \end{ a l i g n ∗}

\noindent where $ K = \ f r a c { u }{ v }$ and $ g = \ l e f t (\ begin { array }{ cc } a & b \\ c & d \end{ array }\ right ) . $ \ h f i l l However if we consider points ( 6 . 3 ) of the infinitesimal cycle

\noindent $ from ˆ{ then } K { y = } v { 0 } u ˆ{ 0 + ve { 1 } . I t a l s o } { = l i n e −two ˆ{\ varepsilon u } { v { 0 } u }{ ue } = \ varepsilon {\underline {\}} { . } Thus } c o i n c i d e s ˆ{ the value } o f { ( } the { up to } l i m i t { an infinitesimal } ( 6 . 5 )$at$the { number }ˆ{ infinitesimal } { ) } { with } s c a l e { the value }$ i s $ independent { in ( 5 . 1 0 ) . }$

\noindent Remark 6 . 9 . \quad There is another connection between parabolic function theory and non − standard a n a l y s i s . \quad As was mentioned in Section 2 , the Clifford algebra $ C−lscript ( p )$ \quad corresponds to the set of

\noindent dual numbers $ u + \ varepsilon v $ with $ \ varepsilon ˆ{ 2 } = 0 [ 7 2 , $ Supplement C ] . On the other hand we may consider the set of numbers $ u + \ varepsilon v $ within the non − standard analysis , with $ \ varepsilon $ b eing an infinitesimal . \quad In t h i s

\noindent case $ \ varepsilon ˆ{ 2 }$ is a higher order infinitesimal than $ \ varepsilon $ and effectively can b e treated as 0 at infinitesimal

\noindent s c a l e o f $ \ varepsilon , $ i . e . we again get the dual numbers condition $ \ varepsilon ˆ{ 2 } = 0 . $ \quad This explains why many results of differential calculus can b e naturally deduced within dual numbers framework [ 1 \quad 1 ] .

Infinitesimal cycles are also a convenient t ool for calculations of invariant measures , Jaco − bians , e t c . 34 .... V period V period Kisil \noindenthline34 34 \ h f i l l V . V . K i s i l V . V . Kisil 7 .. Global .. properties \ [ So\ r far u l e we{3em were}{ interested0.4 pt }\ in] individual properties of cycles and local properties of the point space period .. Now we describe some global properties which are related t o the set of cycles as the whole7 period Global properties \noindent7 periodSo far 17 .. we\ Compactificationquad wereGlobal interested\ ofquad R in to individual thep r o power p e r t iof e properties s sigma of cycles and local properties of the point Givingspace Definition . Now 3 period we describe1 of maps someQ and global M we did properties not consider which properly are their related domains t o the and setranges of periodcycles as \noindentthe wholeSo far . we were interested in individual properties of cycles and local properties of the point For example comma the image of open parenthesisσ 0 comma 0 comma 0 comma 1 closing parenthesis in P to the power of 3 comma which i sspace transformed7 . . 1\quad by Q Compactification tNow o the we equation describe 1 = 0 of some commaR global i s properties which are related t o the set of cycles as the wholenotGiving a valid . conic Definition section in3 .R 1to of the maps power ofQ sigmaand periodM we We did also not did consider not investigate properly yet accuratelytheir domains singular and points of the ranges . M o-dieresis bius map open parenthesis 2 period 33 closing parenthesis period .. It turns out that both questions are connected period \noindentOneFor of the example7 standard . 1 \ ,quad the approaches imageCompactification of .. open(0, 0, 0 square, 1) ∈ P bracket of, which $ 6 R 0 i commaˆ s{\ transformedsigma .. S ..}$ 1 byclosingQ t square o the bracket equation .. to1 deal = 0, withi singularities of the M s dieresis-o bius map i s σ \noindentt onot consider aGiving valid proj conicective Definition coordinates section in 3 onR . the. 1We plane of also maps period did $ .. not Q Since $ investigate we and have $M$ already yet accurately a we projective did not singular space consider of points properly of their domains and ranges . cyclesthe comma .. we may use it as a model for compactification which i s even more appropriate period .. The \noindentidentificationM o¨ biusForexample,theimageof of map points ( 2 with . 3 zero ) . hyphen It turns radius out cycles $( that plays 0both an , questions important 0 , r are o-circumflex 0 connected , le 1 here . ) period\ in P ˆ{ 3 } , $ which i s transformed by $Q$Definition totheequationOne .... of 7 the period standard 1 period $1 approaches .... The = only 0 irregular [ 6 ,$ 0 , point is§ ....1 open ] parenthesis to deal with 0 comma singularities 0 comma of 0 comma the M 1o¨ closing parenthesis in P to the powerbius of map 3 .... ofi s the t map o consider Q is called proj zero ective hyphen coordinates radius on the plane . Since we have already a \noindentcycleprojective at infinitynot aspace and valid denoted of conic cycles by Z sub ,section infinity we may in period use $ R it ˆ as{\ asigma model} for. compactification $ We also did which not investigate i s even yet accurately singular points of the Themore following appropriate results are . easily The obtained identification by direct calculations of points with even withoutzero - radius a computer cycles : plays an important \noindentr oˆ leM here $ .\ddot{o} $ biusmap(2 . 3) . \quad It turns out that both questions are connected . Lemma 7 period 2 period 3 1 periodDefinition Z sub infinity 7 . 1 .. . is theThe image only of irregular the zero hyphen point (0 radius, 0, 0, 1) cycle∈ Z subP of open the parenthesis map Q is 0 called commazero 0 closing parenthesis = open parenthesisOne of- radius the 1 comma standard 0 comma approaches 0 comma 0\ closingquad parenthesis[ 6 0 , \ ..quad at the\S origin\quad under1 reflection ] \quad to deal with singularities of the M $ \openddotcycle parenthesis{o} at$ infinity bius inversion mapand closing i denoted s parenthesis by Z∞ ... into the unit cycle open parenthesis 1 comma 0 comma 0 comma minus 1 closing parenthesis commat o consider sThe ee b following lue cycles proj in results ective Fig period are coordinates easily .. 9 open obtained parenthesis on bythe bdirect planeclosing calculations parenthesis . \quad Since endash even without openwe have parenthesis a already computer d closing a : projective parenthesis period space of c y2 c period lLemma e s , ..\ Thequad 7 following .we 2 . may statements use it are as equivalent a model for compactification which i s even more appropriate . \quad The identificationopen parenthesis1.Z a∞ of closing pointsis the parenthesis image with of .. zero Athe point− zeroradius open - radius parenthesis cycles cycle u plays commaZ(0,0) = an v (1 closing important, 0, 0, 0) parenthesisat rthe $ originin\ Rhat to{ under theo} power$ l e of sigmahere .. . belongs to the zeroreflection hyphen radius cycle Z sub open parenthesis 0 comma 0 closing parenthesis .. centred at the origin semicolon \noindentopen( inversion parenthesisD e f i n) bi t closingi ointo n \ h parenthesis the f i l unit l 7 cycle . .. 1 The . \(1 zeroh, 0 f, i0 hyphenl, l−1)The, s radius ee only b lue cycle irregular cycles Z sub in open Figpoint parenthesis . \ h9 f i ( l ul b comma )$ – ( d v ) closing0. , parenthesis 0 , .. 0is sigma , 1 ) \ in P ˆ{ 3 }$ 2\ .h f i l l Theof following the map statements $Q$ is are called equivalent zero − r a d i u s hyphen orthogonal to zero hyphen radiusσ cycle Z sub open parenthesis 0 comma 0 closing parenthesis semicolon open( a parenthesis) A point c closing(u, vparenthesis) ∈ R belongs .. The inversion to the zero z arrowright-mapsto - radius cycle 1 dividedZ(0,0) bycentred z .. in at the the unit origin cycle is ; singular in the point open \noindent cycle at infinity and denoted by $ Z {\ infty } . $ parenthesis( b u ) comma The v closing zero - parenthesis radius cycle semicolonZ(u,v) is σ− orthogonal to zero - radius cycle Z(0,0); open parenthesis( c ) d closing The inversion parenthesis ..z The7→ 1 imagein of the Z unitsub open cycle parenthesis is singular u comma in the v point closing parenthesis(u, v); .. under inversion in the unit \ centerline {The following results arez easily obtained by direct calculations even without a computer : } cycle is orthogonal( d ) to The Z sub image infinity of periodZ(u,v) under inversion in the unit cycle is orthogonal to Z∞. If any fromIf the any above from is true the we above also issay true that we image also of say open that parenthesis image u of comma(u, v v) closingunder parenthesis inversion .. inunder the inversion in the unit \noindentcycleunit belongsLemma to zero 7 hyphen. 2 . radius cycle at infinity period In the elliptic case the compactificationcycle belongs i s done to zero by addition - radius t o cycle R to the at powerinfinity of e . a point infinity at infinity comma which \ hspace ∗{\ f i l l } $ 1 . Z {\ infty }$ \quad is the image of the zero − radius cycle $ Z { ( 0 , i s theIn elliptic the zero elliptic hyphen case radius the cycle compactification period .. However i s donein the byparabolic addition and thyperbolic o Re apoint cases the∞ singularityat infinity 0of ) the,} which M=(1,0,0,0)$ o-dieresis i s the bius elliptic transform zero i s - not radius lo calised cycle in . a single However point\quad endash in theat the parabolic the denominator origin and under is hyperbolic a zero reflection divisor cases for thethe whole singularity zero hyphen of radius the M cycleo¨ bius period transform .. Thus in i each s not EPH lo calised case the in correct a single compactification point – the i denominator s made by the \ centerlinecupis R a to zero the{( power divisor inversion of sigma for the cup ) \Zquad whole sub infinityintotheunitcycle zero - period radius cycle . $( Thus 1 in each , EPH 0 ,case 0 the correct , − 1 ) ,$ seeblue cycles inFig . \quad 9 ( b ) −− ( d ) . } It iscompactification common t o identify i s the made compactification by the R-dotaccent to the power of e of the space R to the power of e with a Riemann sphere period \ centerlineThis model{ can2 . be\ visualisedquad The by following the stereographic statements proj ection are comma equivalent see open square} bracket 9 comma S .. 1 8 period 1 period 4 closing square bracket .. and Fig period .. 1 3 open parenthesis a closing parenthesis period \ centerline {( a ) \quad Apoint $( uσ , v ) \ in R ˆ{\sigma }$ \quad belongs to the zero − radius cycle A similar model can be provided for the parabolic and∪R hyperbolic∪ Z∞. spaces as well comma see open square bracket 2 5 closing square bracket ..$ and Z { ( 0 , 0 ) }$ \quad centred at the origin ; } Fig periodIt is .. common 1 3 open t parenthesis o identify b the closing compactification parenthesis commaR˙ e of open the parenthesis space Re cwith closing a Riemann parenthesis sphere period .Indeed the space R to the power\ centerline ofThis sigma model{ can( b be can ) identified\ bequad visualisedThe with a zero corresp by− the ondingradius stereographic surface cycle of the proj $ constantZ ection{ ( , see u [ 9 , , § v1 8 ) . 1}$ . 4\ ]quad andi s $ \sigma − $ orthogonal to zero − radius cycle $ Zcurvature{Fig( . : 0 .. 1 the 3 , ( sphere a ) 0 . open A ) similar} parenthesis; $ model} sigma can = be minus provided 1 closing for parenthesis the parabolic comma and the hyperbolic cylinder open spaces parenthesis sigma = 0 closing parenthesisas well comma , see .. [or 2 the 5 ] one hyphen and Fig sheet . hyperboloid 1 3 ( b )open , ( parenthesis c ) . Indeed sigma the = space 1 closingRσ parenthesiscan be identified period \ centerlineThewith map aof corresp{ a( surface c ) onding\ tquad o R toThe surface the power inversion of of the sigma constant i $ s givenz curvature\ bymapsto the polar :\ projf r a the c ection{ sphere1 }{ commaz (σ} see$= − open\1)quad, the squarein cylinder bracket the unit 2 .. 5 cycle comma is Fig singular period in the point 1$ closing ((σ u square= 0) , bracketor v the and ) one Fig ; - sheet $period} hyperboloid 1 3 open parenthesis(σ = 1) a. closingThe map parenthesis of a surface endash t open o R parenthesisσ i s given c by closing the parenthesis period Thesepolar .. surfaces proj ection provide , see .. quotedblleft [ 2 5 , Fig compact . 1 ] andquotedblright Fig . 1 3 .. ( model a ) – of ( c the ) . .. Thesecorresp onding surfaces R to providethe power of sigma .. in the sense that\ centerline ..“ M compact dieresis-o{( d” bius ) \ modelquad The of the image corresp of $ onding Z { (Rσ uin the , sense v that ) }$ \ Mquado¨ biusunder inversion in the unit cycle is orthogonal to $ Ztransformations{\transformationsinfty } which. which$ are lift} areed from lift edR to from the powerRσ by of the sigma proj by ection the proj are ection not are singular not singular on these on these surfaces surfaces period However. the hyperbolic case has it s own caveats which may be easily oversight as in the paper \ hspacecit ed∗{\ aboveHoweverf i comma l l } If the for any hyperbolic example from period the case above .. Ahas compactification itis s true own caveats we of also the which hyperbolic say maythat space be image easily R to of theoversight $ power ( of as u h in by ,the a light v cone )open $ \quad parenthesisunder inversion in the unit which paper \ centerlinethecit hyperbolic ed above{ cycle zero , hyphen for belongs example radius to cycle . zero closingA− compactificationradius parenthesis cycle at infinity of at the infinity will hyperbolic indeed . produce} space a closedRh by M a dieresis-o light cone bius invariant obj ect period However( which it will the not hyperbolic b e satisfactory zero for - radius some other cycle reasons ) at explainedinfinity will in the indeed next subsection produce perioda closed M o¨ bius In theinvariant elliptic obj case ect . the However compactification it will not b e i satisfactory s done by additionfor some other t o reasons $ R ˆ{ explainede }$ a point in $ \ infty $ at infinity , which i sthe the next elliptic subsection zero .− radius cycle . \quad However in the parabolic and hyperbolic cases the singularity o f the M $ \ddot{o} $ bius transform i s not lo calised in a single point −− the denominator is a zero divisor for the whole zero − radius cycle . \quad Thus in each EPH case the correct compactification i s made by the

\ begin { a l i g n ∗} \cup R ˆ{\sigma }\cup Z {\ infty } . \end{ a l i g n ∗}

It is common t o identify the compactification $ \dot{R} ˆ{ e }$ of the space $Rˆ{ e }$ with a Riemann sphere . This model can be visualised by the stereographic proj ection , see [ 9 , \S \quad 1 8 . 1 . 4 ] \quad and Fig . \quad 1 3 ( a ) . A similar model can be provided for the parabolic and hyperbolic spaces as well , see [ 2 5 ] \quad and Fig . \quad 13 ( b ) , ( c ) . Indeed the space $Rˆ{\sigma }$ can be identified with a corresp onding surface of the constant curvature : \quad the sphere $ ( \sigma = − 1 ) ,$ thecylinder $( \sigma = 0 ) , $ \quad or the one − sheet hyperboloid $ ( \sigma = 1 ) . $ The map of a surface t o $R ˆ{\sigma }$ i s given by the polar proj ection , see [ 2 \quad 5 , Fig . 1 ] andFig . 13(a) −− ( c ) . These \quad surfaces provide \quad ‘‘ compact ’’ \quad model o f the \quad corresp onding $ R ˆ{\sigma }$ \quad in the sense that \quad M $ \ddot{o} $ bius

\noindent transformations which are lift ed from $ R ˆ{\sigma }$ by the proj ection are not singular on these surfaces .

\ hspace ∗{\ f i l l }However the hyperbolic case has it s own caveats which may be easily oversight as in the paper

\noindent cit ed above , for example . \quad A compactification of the hyperbolic space $ R ˆ{ h }$ by a light cone ( which the hyperbolic zero − radius cycle ) at infinity will indeed produce a closed M $ \ddot{o} $ bius invariant obj ect . However it will not b e satisfactory for some other reasons explained in the next subsection . Erlangen Program at Large hyphen 1 : .... Geometry of Invariants .... 35 \noindenthlineErlangenErlangen Program Program at Large at - Large 1 :− 1 : \ h f i l l GeometryGeometry of of Invariants Invariants \ h f i l l 35 35 open parenthesis a closing parenthesis .. open parenthesis b closing parenthesis .. open parenthesis c closing parenthesis \ [ Figure\ r u l e 13{3em period}{0.4 .. Compactification pt }\ ] of R to the power of sigma and stereographic projections period 7 period 2 .. open parenthesis Non closing parenthesis hyphen invariance of the upper half hyphen plane ( a ) ( b ) ( c ) The important difference b etween the hyperbolic case and theσ two others i s that \ centerlineLemma 7 period{( a 3 ) period\Figurequad .. In( 13 the b . )e lliptic\Compactificationquad and( parabolic c ) } of casesR and the stereographic upper halfplane projections in R to the . power of sigma is preserved by M o-dieresis bius 7 . 2 ( Non ) - invariance of the upper half - plane \ centerlineThe important{ Figure difference 13 . \quad b etweenCompactification the hyperbolic of case $ and R ˆ{\ thesigma two others}$ iand s that stereographic projections . } transformations from SL sub 2 open parenthesis R closing parenthesis period .. However in the hyperbolicσ case any point open parenthesis u commaLemma v closing 7 parenthesis . 3 . ..In with the v e greater lliptic 0 and can parabolic cases the upper halfplane in R is preserved \noindentbe mappedby M 7o to¨ .bius an 2 arbitrary\quad transformations( point Non open ) − parenthesis frominvarianceSL2 u(R to). oftheHowever thepower upper of primein the half comma hyperbolic− plane v to the case power any of point prime( closingu, v) parenthesis .. with v to with v > 0 can the power of prime equal-negationslash 0 period 0 0 0 \noindentThisbe is mapped illustratedThe important to by an Fig arbitrary period difference 3 point : .. any(u coneb, v etween) fromwith the the familyv hyperbolic6= 0 open. parenthesis case 2 and period the 7 closingtwo others parenthesis i s i that s intersecting the both planes This is illustrated by Fig . 3 : any cone from the family ( 2 . 7 ) i s intersecting the \noindentboth planesLemma 7 . 3 . \quad In the e lliptic and parabolic cases the upper halfplane in $ R ˆ{\sigma }$ EE to the0 power of0 prime and PP to the power of prime over a connected curve comma however intersection0 with the plane HH to the power ofis prime preservedEE hasand two branches byMPP over $ period\ddot a connected{o} $ bius curve , however intersection with the plane HH has two transformationsThebranches lack of invariance . The from inlack the $of hyperbolic SL invariance{ 2 case} in has( the many R hyperbolic important ) . case $ consequences\ hasquad manyHowever in important seemingly in the consequences hyperbolic in case any point $ ( udifferent ,seemingly v areas ) comma $ different\quad for example areaswith , : for $ v example> 0 : $ can FigureFigure 14 period 14 . .... EightEight frames frames from from a continuous a continuous transformation transformation from from future future to the to the past past parts parts of the of the light light \noindentcone . be mapped to an arbitrary point $ ( u ˆ{\prime } , v ˆ{\prime } ) $ \quad with $ v ˆ{\prime } cone period h \neGeometryGeometry0 :. R $ to the: powerR is of not h is split not split by the by the real real axis axis into into two two disj disj oint pieces pieces : :there there i s i a s continuous a continuous path openpath parenthesis ( through through the the light light cone cone at at infinity infinity )closing from parenthesis the upper from half the - plane upper t half o the hyphen lower plane which t o does the lower which does not \ hspacecrossnot the∗{\ realf i laxis l } This open is parenthesis illustrated see sin hyphen by Fig like . curve 3 : \ jquad oinedany two sheets cone of from the hyperbola the family in Fig ( period 2 . 7 .. ) 1 5 i open s intersecting parenthesis a the both planes closingcross parenthesis the real closing axis ( parenthesis see sin - like period curve j oined two sheets of the hyperbola in Fig . 1 5 ( a ) ) . \noindentPhysicsPhysics : ....$ There EE: ˆThere is{\ noprime M is o-dieresis no} M$o bius¨ andbius invariant invariant$ PP way ˆ{\ twayprime o separate t o separate}$ quotedblleft over “ a past past connected ” quotedblright and “ futurecurve and ” , .... parts however quotedblleft of intersection future quotedblright with the plane parts$ HH ofthe ˆ{\ the lightprime light }$ has two branches . Thecone lack opencone squareof [ invariance 6 6bracket ] , i . 6 e 6 . closing in there the square is hyperbolic a continuous bracket comma case family i period has of many M e periodo¨ bius important there transformations is a continuous consequences family reversing of in M dieresis-oseemingly the bius transformations differentarrow areas , for example : reversing the arrow  1 −te  of time periodof time .. For . .. exampleFor example comma .. , the thefamily family .. of matrices of matrices .. Row 1 1 minus te sub1 1 Row, t 2 te∈ sub[0, 1∞ 1) . comma t in open square \noindent Figure 14 . \ h f i l l Eight frames from a continuous transformationte1 1 from future to the past parts of the light bracketprovides 0 comma infinity such closing parenthesis .. provides .. such a transformationa transformation period .. Fig . period Fig .. 1 . 4 illustrates 1 4 illustrates this by corresponding this by corresponding images for eight subsequentimages for eight \noindentvalues of tcone period . subsequent values of t. Analysis : .. There is no a possibility t o split L sub 2 open parenthesis R closing parenthesis .. space of function into a direct sum of the Analysis : There is no a possibility t o split L ( ) space of function into a direct sum \noindentHardy typeGeometry space of functions $ : having Rˆ{ anh analytic}$ is extension not split into the2 byR upper the realhalf hyphen axis plane into and two disj oint pieces : there i s a continuous path ( throughof the Hardythe light type cone space at of functions infinity having ) from an the analytic upper extension half − plane into the t upper o the half lower - plane which does not it sand non hyphen trivial complement comma i period e period .. any function from L sub 2 open parenthesis R closing parenthesis has an quotedblleft analytic extension quotedblright into \ centerline { crossit s non the - real trivial axis complement ( see sin , i− . elike . any curve function j oined from twoL2 sheets(R) has an of “ the analytic hyperbola in Fig . \quad 1 5 ( a ) ) . } theextension upper half hyphen ” into plane in the sense of hyperbolic function theory comma see open square bracket 3 7 closing square bracket period \noindent Physicsthe upper : \ halfh f i - l lplaneThere in the is noMsense of $ hyperbolic\ddot{o} function$ bius theory invariant , see way[ 3 7 t ] . o separate ‘‘ past ’’ and \ h f i l l ‘‘ future ’’ parts of the light

\ hspace ∗{\ f i l l } cone [ 6 6 ] , i . e . there is a continuous family ofM $ \ddot{o} $ bius transformations reversing the arrow

\ hspace ∗{\ f i l l } o f time . \quad For \quad example , \quad the family \quad o f matrices \quad $\ l e f t (\ begin { array }{ cc } 1 & − te { 1 }\\ te { 1 } & 1 \end{ array }\ right ) , t \ in [ 0 , \ infty ) $ \quad provides \quad such

a transformation . \quad Fig . \quad 1 4 illustrates this by corresponding images for eight subsequent values of $t .$

\noindent Analysis : \quad There is no a possibility t o split $ L { 2 } ( R ) $ \quad space of function into a direct sum of the Hardy type space of functions having an analytic extension into the upper half − plane and

\ hspace ∗{\ f i l l } i t s non − trivial complement , i . e . \quad any function from $ L { 2 } ( R ) $ has an ‘‘ analytic extension ’’ into

\ centerline { the upper half − plane in the sense of hyperbolic function theory , see [ 3 7 ] . } 36 .... V period V period Kisil \noindenthline36 36 \ h f i l l V . V . K i s i l V . V . Kisil Figure 15 period .... Hyperbolic objects in the double cover of R to the power of h : open parenthesis a closing parenthesis the .... quotedblleft upper\ [ \ r quotedblright u l e {3em}{0.4 .... pt half}\ hyphen] plane semicolon open parenthesis b closing parenthesis the unit circle period

All and R sub minus to the power of h to the power of the ofh to the power of above R to the power of h comma depicted to the power of problemsFigure can by 15 the . toHyperbolic the power objects of be resolved in the double squares cover ACA of toR the: power ( a ) the of prime “ upper sub ” C half to - the plane power ; ( b of ) prime the unit prime to the power of in the to\noindent the powercircle . ofFigure following 15 and . A\ h to f thei l l powerHyperbolic of prime C objects to the power in of the way double prime open cover square of bracket $ R C ˆ{ primeh } prime: A ( to $ the a power ) the of prime\ h f i l l ‘ ‘ upper ’ ’ \ h f i l l h a l f − plane ; ( b ) the unit circle . prime to the power of 3 7 comma S in sub Fig sub period to the power of A period 3 closing square bracket sub 1 5 to the power of period We \ [ All { and R ˆ{ h } { − }}ˆ{ the } o f ˆ{ above } R ˆ{ h } , depicted ˆ{ problems } can{ by } the ˆ{ be } A.3] take 2 copiesthe correspondinglyabove h subproblems period The R subbe plus to theinthe power offollowing h 0 way0 3 . h All h of , depicted canbythe resolved 0 00 andA C [ 00 7, §in Wetake2copiescorrespondingly r e s o l v e dand { squaresR ACA ˆ{\prime }}ˆ{ insquares theACA} {C C ˆ{\prime C0 \primeFig }}.15ˆ{ f o l l o w i n g } and A. ˆThe{\Rprime+ } boundariesR− of these squares are light cones at infinity and we glue R sub plus to the power0A of h and R sub minus to the power of h in such Ca way ˆ{ thatway {\prime }} [ { C {\prime {\prime { A }}}ˆ{\prime \prime }}ˆ{ 3 } 7 , \S { in } { Fig }ˆ{ A . 3 ] } { . }ˆ{ . } { 1 5 } We take 2 copies { correspondinglyh h } { . The } R ˆ{ h } { + }\ ] theboundaries construction iof s invariantthese squares under the are natural light cones action at of the infinity M dieresis-o and we bius glue transformationR+ and R period− in such .... That a way i s achievedthat if the letters A comma B comma C comma D comma E in Fig period 1 5 are identified regardless of the number of primes attachedthe construction t o them period i s .... invariant The corresponding under the model natural through action a stereographic of the M o¨ projectionbius transformation is presented on . That \noindentFigi period s boundaries 1 6 comma compare of these with Fig squares period 1 are 3 open light parenthesis cones c at closing infinity parenthesis and period we glue $ R ˆ{ h } { + }$ and $ R ˆ{ h } { − }$ inFigure suchachieved 16 a wayperiod if that the .. Double letters coverA, B, of C, the D, E hyperbolicin Fig . space 1 5 are comma identified cf period regardless Fig period of 1the 3 open number parenthesis of primes c closing parenthesis period .. The secondattached hyperboloid t o them is shown . as The corresponding model through a stereographic projection is \noindenta bluepresented skeletonthe onperiod construction .. It is attached i s to invariant the first one under along the the light natural cone at action infinity comma of the which M is $ represented\ddot{o} by$ bius transformation . \ h f i l l That i s twoFig red .lines 1 6 period , compare with Fig . 1 3 ( c ) . \noindentin ThisFigure R-tildewideachieved 16 . toDouble the if power the cover letters of of h the consists hyperbolic $A to the space power , , B cf of . aggregate Fig , . 1 C 3 sub( c ), of . the D The to thesecond , power E$ hyperboloid of denoted in Fig is sub shown .1 upper 5 are to the identified power of by sub regardless of the number of primes halfplaneas ato blue the powerskeleton of . tildewide-R It is attached to the to power the first of h one i s a along 2 in thesub light R sub cone plus at to infinity the power , which of h is to represented the power of by hyphen fold sub and cover of\noindent the subtwo lower redattached lines R to . the power t o them of h period . \ h The f i l l 1The in R to corresponding the power of h period model A throughfrom minus a to stereographic hyperbolic quotedblleft projection upper quotedblright is presented on thicksim half hyphen plane conformally \noindent Fig . 1 6 , compare with Fig . 1 3 ( c ) . invariant twoaggregate hyphendenoted foldby coverRehisa of the Minkowski−fold space hyphenh t ime wash constructed− in open square00 bracket 6 6 comma S III period 4 inThis h ofthe upperhalfplane2in h andcoverofthelowerR . The1inR . Ahyperbolic“upper ∼half−planeconformally closing squareRe bracketconsists in connection R+ \noindentwith the redFigure shift problem 16 . \ inquad extragalacticDouble astronomy cover of period the hyperbolic space , cf . Fig . 1 3 ( c ) . \quad The second hyperboloid is shown as aRemark blueinvariant skeleton 7 period two 4 period- . fold\quad coverIt of is the attached Minkowski to space the first- t ime one was along constructed the light in [ 6 6 cone , §III at . 4 infinity ] in , which is represented by two1 periodconnection red .. lines The hyperbolic .with the orbit red ofshift the problem K subgroup in in extragalactic the R-tildewide astronomy to the power . of h consists of two branches of the hyperbola passingRemark through 7 open . 4 parenthesis . 0 comma v closing parenthesis in R sub plus to the power of h and open parenthesis 0 comma minus v to h the\ [ power in of This1 minus .{\ 1 The closingwidetilde hyperbolic parenthesis{R} orbit inˆ{ Rh of sub} the minusc oK nsubgroups to i s the t s power}ˆ{ inaggregate of the h commaRe consists see} { Figoof periodf two the .. branches 1} 5ˆ period{ denoted of .. the If we} watch{ upper the rotation}ˆ{ by } { h a l f p l a n e }ˆ{\ widetilde {R} ˆ{ h } iof sahyperbola straight a } line2{ generatingin }ˆ{ − a conef o ..l d open} { parenthesisR ˆ{ h } 2 period{ + }} 7 closing{ and parenthesis} cover .. then o f it s{ intersectionthe } { withlower the} planeR HHˆ{ toh the} power. The { 1 } in R ˆ{ h } . A ˆ{ − } { h y p e rh b o l i c } ‘− ‘1 upperh ’ ’ {\sim } h a l f − plane { conformally }\ ] of prime .. on passing through (0, v) in R+ and (0, −v ) in R−, see Fig . 1 5 . If we watch the Figrotation period 3 open parenthesis d closing parenthesis will draw the both branches period .. As mentioned in Remark 2 period 7 period 2 they 0 have the sameof a straight line generating a cone ( 2 . 7 ) then it s intersection with the plane HH \noindentfocalon length Figinvariant period. 3 ( d ) will two draw− fold the both cover branches of the . Minkowski As mentioned space in− Remarkt ime was 2 . constructed7 . 2 they have in [ 6 6 , \S III . 4 ] in connection with2 periodthe the same .. red The focal quotedblleft shift length problem upper . quotedblrightin extragalactic halfplane astronomy i s bounded by . two disj oint .. quotedblleft real quotedblright .. axes denoted by AA to the power2 . of The prime “ andupper C to ” the halfplane power of i prime s bounded C to the by power two disj of prime oint prime “ real ” axes denoted by 0 0 00 \noindentin FigAA periodandRemark 1C 5C period 7 . 4 . For the hyperbolic Cayley transform in the next subsectionin Fig . 1 we 5 need . the conformal version of \ hspacethe hyperbolic∗{\Forfthe i l l unit} hyperbolic1 disk . \quad period CayleyThe .. We hyperbolic define transform it in R-tildewide in orbit the next of to the thesubsection power $ K of $ weh as subgroup need follows the : conformal in the $ version\ widetilde {R} ˆ{ h }$ consists of two branches of the hyperbola D-tildewideof = open brace open parenthesis ue sub 0 plus ve sub 1 closing parenthesis bar u to the power of 2 minus v to the power of 2 h greater\ hspacethe minus∗{\ hyperbolic 1f i comma l l } passingthrough u unit in R disksub plus . to We the $( power define of0 it h inclosing ,Re v braceas follows )$ cup open in: brace $Rˆ open{ parenthesish } { + } ue$ sub and 0 plus $ ve ( sub 0 1 closing , parenthesis− v ˆ{ − bar1 } u to) the $ power in $of R 2 minus ˆ{ h v} to{ the − power } , of $ 2 less see minus Fig 1 . comma\quad u1 in 5 R sub. \quad minus toIf the we power watch of h the closing rotation brace period

2 2 h 2 2 h of a straightDe = { line(ue0 + generatingve1) | u − av cone> −1\,quad u ∈ R(+} 2 ∪ . {(ue 70 )+\vequad1) |thenu − itv < s− intersection1, u ∈ R−}. with the plane $ HH ˆ{\prime }$ \quad on Fig . 3 ( d ) will draw the both branches . \quad As mentioned in Remark 2 . 7 . 2 they have the same focal length .

\ hspace ∗{\ f i l l }2 . \quad The ‘‘ upper ’’ halfplane i s bounded by two disj oint \quad ‘ ‘ r e a l ’ ’ \quad axes denoted by $ AA ˆ{\prime }$ and $ C ˆ{\prime } C ˆ{\prime \prime }$

\ centerline { in Fig . 1 5 . }

\ hspace ∗{\ f i l l }For the hyperbolic Cayley transform in the next subsection we need the conformal version of

\noindent the hyperbolic unit disk . \quad We define it in $ \ widetilde {R} ˆ{ h }$ as follows :

\ [ \ widetilde {D} = \{ ( ue { 0 } + ve { 1 } ) \mid u ˆ{ 2 } − v ˆ{ 2 } > − 1 , u \ in R ˆ{ h } { + }\}\cup \{ ( ue { 0 } + ve { 1 } ) \mid u ˆ{ 2 } − v ˆ{ 2 } < − 1 , u \ in R ˆ{ h } { − } \} . \ ] Erlangen Program at Large hyphen 1 : .... Geometry of Invariants .... 37 \noindenthlineErlangenErlangen Program Program at Large at - Large 1 :− 1 : \ h f i l l GeometryGeometry of of Invariants Invariants \ h f i l l 37 37 It can circles b in e R to the power of h from minus to shown R sub plus to the power of h and to the power of that to the power of D-tildewide\ [ \ r u l e { period3em}{ is0.4 conformally pt }\ ] We call tildewide-T the .... invariant open parenthesis and conformal closing parenthesis has unit a boundary circle in R to the power of h period T-tildewide endash 2 Fig sub period 1 5 copies open parenthesis b closing parenthesis illustrates to the Itcancirclesb e h− De . isconformallyWe call T the invariant(andconformal)has aboundarycircle power of of the unitin theR shownthat e unit h and \noindent $ I t can { R+c i r c l e s } b { in }$ e $ R ˆ{ h }ˆ{ − } { shown { R ˆ{ h } { + } and }ˆ{ that }}ˆ{\ widetilde {D}} geometryh of the conformal unit disk in R-tildewideoftheunit to the power of h in comparison with the quotedblleft upper quotedblright half hyphen . isin R conformally.Te − −2Fig. 15{copies(We }$b )illustrates c a l l $ \ widetildethe {T} $ the \ h f i l l $ i n v a r i a n t { ( } and{ conformal } ) plane period h has8 ..{geometry Theunit .. Cayley} ofa the .. transform boundaryconformal .. and{ unitc .. i ther disk c l ..e inunit}$Re .. in cyclein comparison $ R ˆ{ h } with. the\ widetilde “ upper ”{ halfT} - plane −− .2 { Fig } { . 1 5 } c o p i e s { ( }$ bThe $8 ) upper illustrates The half hyphen Cayley plane ˆ{ o is f the universal the transform unit starting}$ point the and for an analytic the function unit theory of cycle any EPH typThe e period upper .. However half - planeuniversal is models the universal are rarely starting b est suited point to particular for an analytic circumstances function period theory .. For ofmany any \noindentreasonsEPH it igeometry typ s more e . convenient ofHowever the t o conformal consideruniversal analytic models unit functions disk are rarely in in the $ b\ unit estwidetilde disk suited rather to{R} thanparticularˆ{ inh the}$ circumstances in comparison with the ‘‘ upper ’’ half − plane . upper. half For hyphen many plane reasons comma it although i s more both convenient theories are t ocompletely consider isomorphic analytic functionscomma of course in the period unit This disk i somorphism \noindenti s deliveredrather8 than by\quad the in .. theThe Cayley upper\quad transform halfCayley -period plane\quad .. , It although s drawbacktransform both i s that\ theoriesquad thereand i sare no\quadcompletely a .. quotedblleftthe \quad isomorphic universalunit unit\ ,quad of diskc quotedblright y c l e comma in eachcourse EPH . case This we i obtain somorphism something i sps delivered ecific from by the the same upperCayley half transform hyphen plane. period It s drawback i s \noindent8 periodthat 1thereThe .. Elliptic upperi s no and a half hyperbolic “− universalplane Cayley unit is transforms the disk universal ” , in each starting EPH case point we obtain for somethingan analytic sp ecific function theory of any EPH typIn ....from e . the\ thequad .... same ellipticHowever upper .... and universal half hyperbolic - plane ....models . cases ....are open rarely square b bracket est suited 3 7 closing to square particular bracket ....circumstances the .... Cayley transform. \quad iFor s .... many givenreasons ....8 .by 1 theit.... i Elliptic s matrix more .... andconvenient C = hyperbolic t o consider Cayley transforms analytic functions in the unit disk rather than in the upperRowIn 1 h 1 the a minus l f − elliptic eplane sub 1 and Row , although hyperbolic 2 sigma e sub both cases 1 1 theories . comma [ 3 7 ] where the are sigmaCayley completely = transform2 e sub isomorphic 1 open i s parenthesis given , by of 2 the periodcourse matrix 1 closing . This parenthesis i somorphism and det C i sC delivered= by the \quad Cayley transform . \quad It s drawback i s that there i s no a \quad ‘‘ universal unit disk ’’ , = 2 period It can be applied as the M dieresis-o bius transformation in each1 EPH−e case1 we obtain something sp ecific from the same upper half − plane . Equation: open parenthesis, where 8σ period= 2e1 (2 1.1) closingand parenthesisdet C = 2. ..RowIt can 1 1 be minus applied e sub as 1 theRow M 2 sigmao¨ bius e sub transforma- 1 1 . : w = open parenthesis ue sub σe1 1 0 plus vetion sub 1 closing parenthesis arrowright-mapsto Cw = open parenthesis ue sub 0 plus ve sub 1 closing parenthesis minus e sub 1 divided by\noindent sigma e sub8 1 . open 1 \ parenthesisquad Elliptic ue sub 0 and plus hyperbolic ve sub 1 closing Cayley parenthesis transforms plus 1 t o a point open parenthesis ue sub 0 plus ve sub 1 closing parenthesis in R to the power of sigma period .... Alternatively it acts by \noindent In \ h f i l l the \ h f i l l e l l i p t i c \ h f i l l and hyperbolic \ h f i l l c a s e s \ h f i l l [ 3 7 ] \ h f i l l the \ h f i l l Cayley transform i s \ h f i l l given \ h f i l l by the \ h f i l l matrix \ h f i l l conjugation g C = 1 divided by1 2 CgC−e to1 the power of minus 1 on an element(ue0 + ve1) − e1 $ CLine = 1 g $ in SL sub 2 open parenthesis R closing: w = parenthesis (ue0 + ve1) :7→ LineCw 2= g C = 1 divided by 2 Row 1 1 minus(8 e.1) sub 1 Row 2 sigma e sub 1 σe1 1 σe1(ue0 + ve1) + 1 1 . Row 1 a be sub 0 Row 2 minus ce sub 0 d . Row 1 1 e sub 1 Row 2 minus sigma e sub 1 1 . period open parenthesis 8 period 2 closing parenthesis\noindent $\ l e f t (\ begin { arrayσ }{ cc } 1 & − e { 1 }\\\sigma 1 e {−1 1 } & 1 \end{ array }\ right ) , $ t o a point (ue0 + ve1) ∈ R . Alternatively it acts by conjugation gC = 2 CgC on an element whereThe .. $ connection\sigma between= 2 the{ twoe ..{ forms1 }} .. open( parenthesis 2 . 8 1 period )$ 1 closing anddet parenthesis $C .. and = .. 2 open .$ parenthesis ItcanbeappliedastheM 8 period 2 closing $ \ddot{o} $ bius transformation parenthesis .. of the Cayley transform i s .. given by g ∈ SL2(R): g C to the power of Cw = C open1  parenthesis1 −e  gw closing  a parenthesis be  comma1 i periode  e period C intertwines the actions of g and g C period \ begin { a l i g n ∗} gC = 1 0 1 . (8.2) The Cayley transform open parenthesis2 σe1 u to1 the power−ce of0 primed e sub 0− plusσe1 v to1 the power of prime e sub 1 closing parenthesis = C open parenthesis\ l e f t (\ begin ue sub{ array 0 plus}{ vecc sub} 1 closing & − parenthesise { 1 in}\\\ the ellipticsigma case i se very{ important1 } & 1open\end square{ array bracket}\ right 5 5 comma) : w = ( ue S{ IXThe0 period} + connection 3 closing ve { square1 between} bracket) the\ commamapsto two open formsCw square = ( bracket 8\ f . r a 1 c 6 ){ 8( comma and ue Chapter{ ( 80 .} 2 8 )+ comma of ve theopen{ Cayley1 parenthesis} ) − 1 periode { 1 21 closing}}{\sigma parenthesise { transform1 } closing( iue square s{ given bracket0 } by+ bothgC veCw for= complex{C(1gw}), i analysis) . e +.C andintertwines 1 representation}\ tag ∗{ the$ ( actions theory 8 of of . SLg suband 1 2gC. ) open $} parenthesis R closing parenthesis \end{ a l i g n ∗} 0 0 period The Cayley transform (u e0 + v e1) = C(ue0 + ve1) in the elliptic case i s very important [ 5 5 , The§ transformationIX . 3 ] , [ 6 8 g ,arrowright-mapsto Chapter 8 , ( 1 g . C 1 open 2 ) ] parenthesis both for complex8 period 2 analysis closing parenthesis and representation i s an isomorphism theory of the groups SL sub 2 open \noindent toapoint $( ue { 0 } + ve { 1 } ) \ in R ˆ{\sigma } . $ \ h f i l l Alternatively it acts by conjugation parenthesisof SL R2 closing(R). The parenthesis transformation and SU openg 7→ parenthesisgC(8.2) i s 1 ancomma isomorphism 1 closing parenthesis of the groups namelySL2(R) and SU ( $ gin C-lscript1 C , 1 ) = namely open\ f r parenthesis a c { in 1C}{− elscript2 closing} (eCgC) parenthesiswe ˆ have{ − we1 have}$ on an element Equation: h = open parenthesis a minus d closing parenthesis e sub 1 plus open parenthesis b plus c closing parenthesis e sub 0 period open parenthesis\ [ \ begin { 8a periodl i g n e d 3} closingg \ parenthesisin SL ..{ g2 C} = 1(R): divided by 2 Row\\ 1 f h Row 2 minus h f . comma with f = open parenthesis a plus d 1  f h  closingg parenthesis C = gC\ minus=f r a c { open1 }{ parenthesis2 ,}\withl e b f t minusf(\=begin (a c+ closingd{)array− ( parenthesisb −}{c)ecce} 1 eand sub & 1− e subhe 0= and{ (a 1− d}\\\)e + (bsigma+ c)e .(8.3)e { 1 } & 1 \end{ array }\ right ) \ l e f t (\ begin { array }{ cc } a 2 −h f 1 0 1 0 &Under be { the0 map}\\ R − to thece power{ 0 of} e right& d arrow\end C{ array open parenthesis}\ right ) 2\ periodl e f t ( 2\ begin closing{ parenthesisarray }{ cc this} 1 matrix & e becomes{ 1 Row}\\ 1 − alpha \ betasigma e { 1 } & 1 \end{ array }\ right ) . ( 8 . 2 ) \end{ a l i g n e d }\ ] Row 2 beta-macron macron-alpha . comma i period e period the standard α form β of elements Under the map e → (2.2) this matrix becomes , i . e . the standard form of of SU open parenthesis 1 commaR 1 closingC parenthesis .. open square bracketβ¯ α¯ 5 5 comma S IX period 1 closing square bracket comma open square bracket 6 8 comma Chapter 8 comma open parenthesis 1 period 1 1 closing parenthesis closing square bracket period \noindentelementsThe \quad connection between the two \quad forms \quad ( 8 . 1 ) \quad and \quad ( 8 . 2 ) \quad of the Cayley transform i s \quad given by Theof images SU ( of 1 elliptic , 1 ) actions [ 5 5 of, § subgroupsIX . 1 ] , A [ 6 comma 8 , Chapter N comma 8 K , ( are 1 .given 1 1 in ) ] Fig . period 1 7 open parenthesis E closing parenthesis period .. The$ g typC es ofˆ{ Cw } = C ( gw ) ,$ i .e $. C$ intertwinestheactionsof $g$ and $g C . $ The images of elliptic actions of subgroups A, N, K are given in Fig . 17(E). The typ es orbitsof can orbits b e easily can b distinguished e easily distinguished by the number by of the fixed number points on of thefixed boundary points : on .. two the comma boundary one and: two zero correspondingly period .. Although a closer insp ection demonstrate that there are always two fixed The Cayley, one and transform zero correspondingly $ ( u ˆ{\ .prime Although} e a{ closer0 } insp+ ection v ˆ{\ demonstrateprime } e that{ 1 there} ) are = C ( ue { 0 } pointsalways comma two either fixed : points , either : +bullet ve ..{ one1 strictly} ) $ inside in and the one elliptic strictly outside case of i the s unit very circle important semicolon [or 5 5 , • one strictly inside and one strictly outside of the unit circle ; or \SbulletIX ... one 3 ] double , [ 6 fixed 8 point, Chapter on the unit 8 , circle ( 1 semicolon . 1 2 ) or ] both for complex analysis and representation theory of • one double fixed point on the unit circle ; or $ SLbullet{ ..2 two} different( R fixed ) points . $ exactly on the circle period The transformation $• g two\mapsto differentg fixed C points ( exactly 8 . on 2 the circle )$ . i sanisomorphismofthegroups $SL { 2 } ( R )$ andSU(1 ,1)namely in $ C−lscript ( e )$ wehave

\ begin { a l i g n ∗} g C = \ f r a c { 1 }{ 2 }\ l e f t (\ begin { array }{ cc } f & h \\ − h & f \end{ array }\ right ) , with f = ( a + d ) − ( b − c ) e { 1 } e { 0 } and \ tag ∗{$ h = ( a − d ) e { 1 } + ( b + c ) e { 0 } . ( 8 . 3 ) $} \end{ a l i g n ∗}

\ hspace ∗{\ f i l l }Under the map $ R ˆ{ e }\rightarrow C ( 2 . 2 )$ thismatrixbecomes $\ l e f t (\ begin { array }{ cc }\alpha & \beta \\\bar{\beta} & \bar{\alpha}\end{ array }\ right ) , $ i . e . the standard form of elements

\noindent o f SU ( 1 , 1 ) \quad [ 5 5 , \S IX.1] , [68 ,Chapter8 ,(1.11) ] .

Theimages of elliptic actions of subgroups $A , N , K$ are given in Fig $ . 1 7 ( E ) . $ \quad The typ es o f orbits can b e easily distinguished by the number of fixed points on the boundary : \quad two , one and zero correspondingly . \quad Although a closer insp ection demonstrate that there are always two fixed points , either :

\ centerline { $ \ bullet $ \quad one strictly inside and one strictly outside of the unit circle ; or }

\ centerline { $ \ bullet $ \quad one double fixed point on the unit circle ; or }

\ centerline { $ \ bullet $ \quad two different fixed points exactly on the circle . } 38 .... V period V period Kisil \noindenthline38 38 \ h f i l l V . V . K i s i l V . V . Kisil Consideration .... of Fig period .... 1 2 open parenthesis b closing parenthesis .... shows .... that .... the parabolic subgroup .... N i s like .... a\ [ phase\ r u transition l e {3em}{0.4 pt }\ ] b etween the elliptic subgroup K and hyperbolic A comma cf period open parenthesis 1 period 1 closing parenthesis period In someConsideration sense the elliptic of Fig Cayley . transform1 2 ( b ) swaps shows complexities that the : .. parabolic by contract subgroup t o the upperN i s like a phase \noindenthalftransition hyphenConsideration plane the K hyphen\ h actionf i l l o i f s now Fig simple . \ h butf i l l ..1 A and2 ( N b are ) not\ h f period i l l shows .. The\ simplicityh f i l l that of K ..\ orbitsh f i l l i sthe parabolic subgroup \ h f i l l $ Nexplained $b etween i s by l i diagonalisation k the e \ elliptich f i l l subgroupa of phase matrices transitionK : and hyperbolic A, cf . ( 1 . 1 ) . Equation:In some open parenthesis sense the 8 elliptic period 4 Cayley closing parenthesis transform .. swaps 1 divided complexities by 2 Row 1 1: minus by e contract sub 1 Row t 2 o minus the e sub 1 1 . Row 1 cosine phi\noindent minusupper e subb half 0etween sine - plane phi the Row the 2elliptic minusK− action e sub subgroup 0 i sine s now phi cosine simple $K$ phi but .and Row hyperbolicA 1 1and e subN 1are Row $A not 2 e sub. , 1 The $1 . = cf simplicity parenleftbigg . ( 1 . 1 to ) the . power of epsilon sub 0 toof theK powerorbits of i iphi s epsilon explained to the by power diagonalisation of 0 i phi parenrightbigg of matrices comma : Inwhere some i sense= e sub the 1 e sub elliptic 0 behaves Cayley as the complex transform imaginary swaps unit complexities comma i period e : period\quad i toby the contract power of 2 t= minuso the 1 upper period h a l f − plane the $ K − $ action i s now simple but \quad $A$ and $N$ are not . \quad The simplicity of A hyperbolic version1 of the1 Cayley−e transform  cos wasφ used− ine opensin φ square  bracket1 e  3 7 closing square bracket period .. The above formula open $ K $ \quad o r b i t s i s 1 0 1 εiφ 0iφ parenthesis 8 period 2 closing parenthesis in R to the power of h = ( 0 ε ), (8.4) 2 −e1 1 −e0 sin φ cos φ e1 1 explainedb ecomes as by follows diagonalisation : of matrices : 2 Equation:where open i = e parenthesis1e0 behaves 8 period as the 5 closing complex parenthesis imaginary .. g C unit = 1 , divided i . e . byi = 2 Row−1. 1 f h Row 2 h f . comma with f = a plus d minus open parenthesis\ begin { aA l b i hyperbolicg plus n ∗} c closing version parenthesis of the e sub Cayley 1 e sub transform 0 and h = open was parenthesisused in [ 3 d 7 minus ] . a closing The above parenthesis formula e sub ( 1 plus open parenthesis b \ f r a c { 1 }{ 2 }\ l e f t (\ begin { array }{ cc } 1 & − e { 1 }\\ − e { 1 } & 1 \end{ array }\ right ) \ l e f t (\ begin { array }{ cc }\cos minus c8 closing . 2 ) in parenthesisRh b ecomes e sub 0 as comma follows : \phiwith& some− subtlee differences{ 0 }\ insin comparison\phi with\\ open − parenthesise { 0 }\ 8 periodsin 3 closing\phi parenthesis& \cos period\phi .... The\end corresp{ array onding}\ right A comma) \ l e N f t (\ begin { array }{ cc } 1 &and eK orbits{ 1 }\\ e { 1 } & 1 \end{ array }\ right ) = ( ˆ{\ varepsilon }ˆ{ i \phi } { 0 }\ varepsilon ˆ{ 0 { i   \phiare}} given), on Fig period\ tag1∗{ 1$ 7f open ( h parenthesis 8 . 4 H closing ) $} parenthesis period .. However there i s an important distinction b etween the elliptic gC = , with f = a + d − (b + c)e1e0 and h = (d − a)e1 + (b − c)e0, (8.5) and\end{ a l i g n ∗} 2 h f hyperbolic cases similar t o one discussed in Subsection 7 period 2 period \noindentLemmawith 8 some periodwhere subtle 1 iperiod $ differences = e { in1 comparison} e { 0 } with$ behaves ( 8 . 3 ) as. the The complex corresp onding imaginaryA, N and unitK , i . e $ . i ˆ{ 2 } = 1− periodorbits1 .. In . the $ elliptic case the .. quotedblleft real axis quotedblright .. U is transformed to the unit circle and the upper half hyphen planeare endash given to on the Fig unit. disk17( :H). However there i s an important distinction b etween the elliptic A hyperbolicEquation:and hyperbolic open version parenthesis cases of 8 similar the period Cayley 6 t closing o one transform parenthesis discussed was ..in open Subsection used brace in open [ 7 3. parenthesis 2 7 . ] . \quad u commaThe v aboveclosing parenthesis formula ( bar 8 v . = 2 0 )closing in brace$ R ˆ rightLemma{ h arrow}$ 8 open . 1 brace . open parenthesis u to the power of prime comma v to the power of prime closing parenthesis bar l sub c sub e tob the ecomes power1 . of as 2 openfollows In the parenthesiselliptic : ucase to thethe power of “ prime real axis e sub ” 0 plusU vis to transformed the power of prime to the e unitsub 1 circle closing and parenthesis = u to the power of primethe 2 plusupper v to half the - power plane of –prime to the 2 = unit 1 closing disk brace: comma Equation: open parenthesis 8 period 7 closing parenthesis .. open brace open\ begin parenthesis{ a l i g n ∗} u comma v closing parenthesis bar v greater 0 closing brace right arrow open brace open parenthesis u to the power of prime g C = \ f r a c { 1 }{ 2 }\ l e f t (\ begin { array }{ cc } f & h \\ h & f \end{ array }\ right ) , with f comma v to the power of prime closing parenthesis bar0 l0 sub c sub2 e0 to the0 power of02 2 open02 parenthesis u to the power of prime e sub 0 plus v = a + d − {((u, v) b| +v = c0} → ) {(u , ve ) { | 1 l} (uee0 +{ v0e1}) = uand+ v h= 1} =, ( d (8−.6) a ) e { 1 } + to the power of prime e sub 1 closing parenthesis = u to the powerce of prime 2 plus v to the power of prime 2 less 1 closing brace comma ( b − c ){ e(u, v{) 0 |} v, > \0}tag →∗{ {(u$0, v(0) 8| l2 .(u0e 5+ v0e )) $=}u02 + v02 < 1}, (8.7) where the length from centre l sub c sub e to the power of 2ce .. is0 given by1 .. open parenthesis 5 period 5 closing parenthesis for sigma = sigma-breve\end{ a l i g n =∗} minus 1 period .. On both s ets SL sub 2 open parenthesis R closing parenthesis 2 acts transitivelywhere the and length the unit from circle centre is generatedlc is comma given for by example( 5 comma . 5 ) for .. by theσ = pointσ ˘ = − open1. parenthesisOn both s 0 ets comma 1 closing parenthesis \noindent with some subtle differencese in comparison with ( 8 . 3 ) . \ h f i l l The corresp onding $ A , .. and theSL2(R) N$unit and diskacts is $K$ generated transitively orbits by open and parenthesis the unit circle 0 comma is generated 0 closing parenthesis , for example period , by the point ( 0 , 1 ) 2 periodand the .. In the unit hyperbolic disk is generated case the .. quotedblleft by ( 0 , 0 real ) . axis quotedblright U is transformed to the hyperbolic unit circle : \noindent2Equation: . InaregivenonFig theopenhyperbolic parenthesis 8 case periodthe $. 8 closing 1 “ real parenthesis 7 axis ( ” .. HU openis ) transformed brace .$ open\ parenthesisquad to theHowever hyperbolic u comma there v unit closing i circle s parenthesis an : important bar v = distinction 0 closing b etween the elliptic and bracehyperbolic right arrow cases open brace similar open parenthesis t o one discussedu to the power in of Subsection prime comma v 7 to . the 2 power. of prime closing parenthesis bar l sub c sub h to the power of 2 open parenthesis u to the power of prime comma v to the power of prime closing parenthesis = u to the power of prime 2 minus \noindent Lemma 8 . 1{( .u, v) | v = 0} → {(u0, v0) | l2 (u0, v0) = u02 − v02 = −1}, (8.8) v to the power of prime 2 = minus 1 closing brace comma ch where the length from centre l sub c sub h to the power of 2 .. is given by .. open parenthesis 5 period 5 closing parenthesis for sigma = 1 . \quad In thewhere elliptic the length case from the centre\quad l2‘‘ realis given axis by ’’ \(quad 5 . 5 )$for U $σ = isσ ˘ = transformed 1. On the to the unit circle and the upper half − sigma-breve = 1 period .. On the hyperbolic unit ch planecirclehyperbolic−− SL subto 2 the openunit unit parenthesis disk R : closing parenthesis .. acts transitively and it is generated comma for example comma .. by point open parenthesiscircle 0 commaSL2 1(R closing) acts parenthesis transitively period and SL it sub is generated2 open parenthesis , for example R closing , parenthesis by point (0, 1).SL2(R) \ beginactsacts also{ a l ialso transitively g n ∗}transitively on the whole on the complement whole complement \{open( brace u open , parenthesis v ) u to\mid the powerv of prime = 0 comma\}\ v to therightarrow power of prime\{ closing( parenthesis u ˆ{\ barprime l sub} c sub, h to v the ˆ{\ powerprime of 2} ) \mid l ˆ{ 2 } { c { e }} {((u0, v0) u ˆ|{\l2prime(u0e +}v0e e) 6={−10} } + v ˆ{\prime } e { 1 } ) = u ˆ{\prime open parenthesis u to the power of prime e sub 0 plus v toch the power0 of1 prime e sub 1 closing parenthesis equal-negationslash minus 1 closing 2 } + v ˆ{\prime 2 } = 1 \} , \ tag ∗{$ ( 8 . 6 ) $}\\\{ ( u , v ) \mid brace to the unit circle , i . e . on its “ inner ” and “ outer ” parts together . v to> the unit0 circle\}\ commarightarrow i period e period\{ .. on( its .. u quotedblleft ˆ{\prime inner} quotedblright, v ˆ{\prime and .. quotedblleft} ) \mid outer quotedblrightl ˆ{ 2 } parts{ c together{ e }} ( u ˆ{\Theprime last feature} e { of0 the} hyperbolic+ v ˆ{\ Cayleyprime transform} e { can1 } be) treated = uin ˆ a{\ wayprime describ2 ed} in+ v ˆ{\prime 2 } period the end of Subsection 7 . 2 , see also Fig . 1 5 ( b ) . With such an arrangement < The1 last\} feature, of\ tag the∗{ hyperbolic$ ( 8 Cayley . transform 7 ) can $} be treated in a way describ ed in the \end{thea l i g hyperbolic n ∗} Cayley endtransform of Subsection maps .. 7 period the “ 2 uppercomma ”.. see half also - Figplane period from .. 1 Fig 5 open . 1 parenthesis 5 ( a ) onto b closing the parenthesis “ inner ” period part .. of With such an arrangement the hyperbolicthe unit Cayley disk \ hspacetransform∗{\ mapsf i l l } thewhere .... quotedblleft the length upper from quotedblright centre .... $ half l ˆ hyphen{ 2 } plane{ c from{ e Fig}} period$ \quad .... 1 5i open s given parenthesis by \quad a closing( parenthesis5 . 5 ) f o r $ \sigmafrom Fig= . 1\breve 5 ( b ){\ .sigma} = − 1 . $ \quad On both s ets $ SL { 2 } ( R ) $ onto the ....One quotedblleft may wish inner that quotedblright the hyperbolic .... part Cayley of the transform unit disk diagonalises the action of subgroup A, fromor Fig some period conjugated 1 5 open parenthesis , in a fashion b closing similar parenthesis t o the period elliptic case ( 8 . 4 ) for K. Geometrically actsOne transitively may wish that the and hyperbolic the unit Cayley circle transform is generateddiagonalises the , foraction example of subgroup , \ Aquad commaby the point ( 0 , 1 ) \quad and the unitit disk will correspondis generated to hyperbolic by ( 0 , rotations0 ) . of hyperbolic unit disk around the origin . Since or somethe conjugatedorigin i s comma the image in a fashion of the similar point t oe1 thein elliptic the upper case open half parenthesis - plane under 8 period the 4 closing Cayley parenthesis transform for K period .. Geometrically it will , we will use the \ centerlinecorrespond to{2 hyperbolic . \quad rotationsIn the of hyperbolic hyperbolic unit case disk thearound\quad the origin‘‘ real period axis .. Since ’’ the $ origin U $ i s is transformed to the hyperbolic unit circle : } the image of the point e sub 1 in the upper half hyphen plane under the Cayley transform comma we will use the \ begin { a l i g n ∗} \{ ( u , v ) \mid v = 0 \}\rightarrow \{ ( u ˆ{\prime } , v ˆ{\prime } ) \mid l ˆ{ 2 } { c { h }} ( u ˆ{\prime } , v ˆ{\prime } ) = u ˆ{\prime 2 } − v ˆ{\prime 2 } = − 1 \} , \ tag ∗{$ ( 8 . 8 ) $} \end{ a l i g n ∗}

\ hspace ∗{\ f i l l }where the length from centre $ l ˆ{ 2 } { c { h }}$ \quad i s given by \quad ( 5 . 5 ) f o r $ \sigma = \breve{\sigma} = 1 . $ \quad On the hyperbolic unit

c i r c l e $ SL { 2 } ( R ) $ \quad acts transitively and it is generated , for example , \quad by point $ ( 0 , 1 ) . SL { 2 } ( R ) $ acts also transitively on the whole complement

\ [ \{ ( u ˆ{\prime } , v ˆ{\prime } ) \mid l ˆ{ 2 } { c { h }} ( u ˆ{\prime } e { 0 } + v ˆ{\prime } e { 1 } ) \ne − 1 \}\ ]

\ centerline { to the unit circle , i . e . \quad on i t s \quad ‘‘ inner ’’ and \quad ‘‘ outer ’’ parts together . }

The last feature of the hyperbolic Cayley transform can be treated in a way describ ed in the end of Subsection \quad 7 . 2 , \quad see also Fig . \quad 1 5 ( b ) . \quad With such an arrangement the hyperbolic Cayley

\noindent transform maps the \ h f i l l ‘ ‘ upper ’ ’ \ h f i l l h a l f − plane from Fig . \ h f i l l 1 5 ( a ) onto the \ h f i l l ‘ ‘ inner ’ ’ \ h f i l l part of the unit disk

\noindent from Fig . 15 ( b ) .

One may wish that the hyperbolic Cayley transform diagonalises the action of subgroup $ A , $ or some conjugated , in a fashion similar t o the elliptic case ( 8 . 4 ) for $K . $ \quad Geometrically it will correspond to hyperbolic rotations of hyperbolic unit disk around the origin . \quad Since the origin i s the image of the point $ e { 1 }$ in the upper half − plane under the Cayley transform , we will use the Erlangen Program at Large hyphen 1 : .... Geometry of Invariants .... 39 \noindenthlineErlangenErlangen Program Program at Large at - Large 1 :− 1 : \ h f i l l GeometryGeometry of of Invariants Invariants \ h f i l l 39 39 Figure 1 7 period .. The unit disks and orbits of subgroups A comma N and K : open parenthesis E closing parenthesis : The elliptic unit disk\ [ \ semicolonr u l e {3em open}{0.4 parenthesis pt }\ ] P sub e closing parenthesis comma open parenthesis P sub p closing parenthesis comma open parenthesis P sub h closing parenthesis : .. The elliptic comma parabolic and hyeprbolic flavour of the parabolic unit disk open parenthesisFigure the 1 pure 7 . parabolicThe unit type disks and orbits of subgroups A, N and K :(E) : The elliptic unit disk ; (Pe), (Pp), \noindentopen(P parenthesish) :Figure The elliptic P sub 1 7 p , parabolic closing. \quad parenthesis andThe hyeprbolic unit transform disks flavour is and very of the similar orbits parabolic with of unit Figs subgroups disk period ( the 1and pure $A 2parabolic open , parenthesis type N$(Pp and K) sub $Kp closing : parenthesis ( E ) :$transform The is elliptic very similar unit with diskFigs . 1 $; and 2(K (p)). P(H){ : Thee } hyperbolic),(P unit disk . { p } ) , $ closing parenthesis period open parenthesis H closing parenthesis : The hyperbolic unit disk period $fix ( subgroup P { Ah sub} h)0 to the : power $ \quad of primeThe open elliptic parenthesis1 ,e0 2 parabolic period 9 closing and parenthesis hyeprbolic conjugated flavour t o of A by the Row parabolic 1 1 e sub 0 Rowunit 2 disk e ( the pure parabolic type fix subgroup Ah(2.9) conjugated t o A by ∈ SL2(R). Under the Cayley map ( 8 . sub$ 0 ( 1 . in P SL{ subp 2} open) parenthesis$ transform R closing is parenthesis very similare period0 1 with Under Figs the Cayley . 1 and map open $ 2 parenthesis ( K 8{ periodp } 5 closing)).(H parenthesis the )subgroup :5 $ ) the The A sub hyperbolic h to the power unit of prime disk b . ecame comma cf period open square bracket 3 7 comma open parenthesis 3 period 6 closing 0 parenthesissubgroup commaA openh b parenthesisecame , cf 3 . period [ 3 7 , 7 ( closing 3 . 6 parenthesis ) , ( 3 . 7 closing ) ] : square bracket : \noindent1 divided byfix 2 Row subgroup 1 1 e sub 1$ Row A ˆ{\ 2 minusprime e sub} 1{ 1h . Row} ( 1 hyperpolic 2 . cosine 9 t minus )$ e conjugatedto sub 0 hyperbolic sine $A$ t Row 2 e by sub $0\ hyperbolicl e f t (\ begin { array }{ cc } 1 & e { 0 }\\ e { 0 } & 1 \end{ array }\ right ) \ in SL { 2 } ( R ) . $ Under the Cayleymap( 8 . 5 ) the sine t hyperpolic1  cosine1 te . Row 1 1 minuscosh t e sub−e 1sinh Rowt 2 e sub  1 1 1− .e = Row 1 exponent exp(e e opent) parenthesis 0 e sub 1 e sub 0 t closing parenthesis 0 Row 2 0 exponent open parenthesis1 e sub 1 e sub0 0 t closing parenthesis1 . comma= 1 0 , 2 −e1 1 e0 sinh t cosh t e1 1 0 exp(e1e0t) \noindentwhere expsubgroup open parenthesis $ A e ˆsub{\ 1prime e sub 0 t} closing{ h } parenthesis$ became,cf.[37,(3.6),(3.7)]: = cosh open parenthesis t closing parenthesis plus e sub 1 e sub 0 sinh open parenthesis t closing parenthesis period .. This obviously corresponds t o hyperbolic rotations \ [ \ f r awhere c { 1 exp}{ 2(e1}\e0t)l = e fcosh t (\ begin(t)+e{1earray0 sinh}{(tcc). } This1 & obviously e { 1 corresponds}\\ − e t o{ hyperbolic1 } & 1 rotations\end{ array }\ right ) \ l e f t (\ begin { array }{ cc }\cosh of R to theh power of h period .. Orbits of the fix subgroups0 0 A sub0 h to the power of comma to the power of prime N sub p to the power of t & of− R .e Orbits{ 0 }\ ofsinh the fix subgroupst \\ e {A0h, N}\p andsinhKe fromt & Lemma\cosh 2 . 8t \end under{ array the}\ Cayleyright ) \ l e f t (\ begin { array }{ cc } 1 prime andtransform K sub e to are the shown power on of prime Fig . from 1 Lemma 8 , which 2 period should 8 .. b under e compared the Cayley with transform Fig . 4 . However & are− showne { on1 Fig}\\ periode ..{ 1 81 comma} & which1 \end should{ array b e}\ comparedright ) with = Fig\ l eperiod f t (\ begin .. 4 period{ array .. However}{ cc }\ theexp parabolic( Cayley e { 1 } e { 0 } t )the & parabolic 0 \\ 0 Cayley & \exp ( e { 1 } e { 0 } t ) \end{ array }\ right ), \ ] transformtransform requires requires a separate a separate discussion discussion period .

\noindent where exp $ ( e { 1 } e { 0 } t ) =$cosh$( t ) + e { 1 } e { 0 }$ sinh $ ( t ) . $ \quad This obviously corresponds t o hyperbolic rotations o f $ R ˆ{ h } . $ \quad Orbits of the fix subgroups $ A ˆ{\prime } { h ˆ{ , }} N ˆ{\prime } { p }$ and $ K ˆ{\prime } { e }$ from Lemma 2 . 8 \quad under the Cayley transform are shown on Fig . \quad 1 8 , which should b e compared with Fig . \quad 4 . \quad However the parabolic Cayley

\noindent transform requires a separate discussion . 40 .... V period V period Kisil \noindenthline40 40 \ h f i l l V . V . K i s i l V . V . Kisil Figure 18 period .. Concentric slash confocal orbits of one parametric subgroups comma cf period Fig period .. 4 period \ [ 8\ periodr u l e {3em 2 .. Parabolic}{0.4 pt }\ Cayley] transforms This case b enefits from a bigger variety of choices period .. The first natural attempt to define a Cayley transform canFigure b e taken 18 . fromConcentric the same formula/ confocal .. orbits open parenthesis of one parametric 8 period subgroups 1 closing , parenthesiscf . Fig . ..4 . with the parabolic value sigma = 0 period\ centerline8 .. The. 2{ Figure Parabolic 18 . Cayley\quad Concentric transforms / confocal orbits of one parametric subgroups , cf . Fig . \quad 4 . } correspondingThis case transformation b enefits from defined a bigger by the variety matrix of .... choices Row 1 1 . minus The e sub first 1 Row natural 2 0 1 attempt . .... and defines to define the shifta one unit \noindentdownCayley period8 transform . 2 \quad canParabolic b e taken Cayley from the transforms same formula ( 8 . 1 ) with the parabolic value σ = 0. The However within the framework .. of this .. paper .. a more general version .. of parabolic .. Cayley \noindentcorrespondingThis case transformation b enefits from defined a bigger by the matrix variety of1 choices−e1 . and\quad definesThe the first shift natural one attempt to define a Cayley transform is possible period .. It i s given by the matrix 0 1 transformC sub sigma-breve can b = e Row taken 1 1 minus from e the sub 1same Row 2formula breve-sigma\quad e sub( 1 8 1 . . comma 1 ) \ ..quad wherewith .. sigma-breve the parabolic = minus 1 value comma 0 $ comma\sigma 1 .. = 0unit . $ \quad The and .. detdown C sub . breve-sigma = 1 .. for all sigma-breve sub period open parenthesis 8 period 9 closing parenthesis Here sigma-breve = minus 1 corresp onds t o the parabolic Cayley transform P sub e with the elliptic flavour comma breve-sigma = 1 endash \noindentHowevercorresponding within the transformation framework of this defined paper by the a matrix more general\ h f i l versionl $\ l e f t of(\ begin parabolic{ array }{ cc } 1 & − e { 1 }\\ t oCayley the parabolic transform Cayley transform is possible P sub . h It with i s the given hyperbolic by the flavour matrix comma cf period .. open square bracket 5 .. 1 comma .. S 2 period 60 closing & 1 square\end{ bracketarray period}\ right .. Finally)$ \ h the f i l l and defines the shift one unit  1 −e  parabolic hyphen parabolicC = transform is1 given, where by an upperσ˘ = hyphen−1, 0, triangular1 and matrix det C from= the 1 endfor of all theσ˘ previous(8.9) σ˘ σe˘ 1 σ˘ . \noindentparagraphdown period . 1 Here σ˘ = −1 corresp onds t o the parabolic Cayley transform P with the elliptic flavour , σ˘ = 1 Fig period .. 1 7 presents these transforms in rows open parenthesis P sube closing parenthesis comma open parenthesis P sub p closing – t o the parabolic Cayley transform P with the hyperbolic flavour , cf . [ 5 1 , §2 parenthesisHowever within and open the parenthesis framework P sub\ hquad closingo fparenthesis t hh i s \quad corresppaper ondingly\quad perioda .. more The row general open parenthesis version P\quad sub p closingof parabolic parenthesis\quad Cayley transform. 6 ] . is Finally possible the . parabolic\quad It - parabolic i s given transform by the is matrix given by an upper - triangular matrix almostfrom coincides the end with of Figs theperiod previous 1 open paragraph parenthesis . A sub a closing parenthesis comma 1 open parenthesis N sub a closing parenthesis and 2 open parenthesis K sub p closing parenthesis period .... Consideration of Fig period .... 1 7 .... by columns from Fig . 1 7 presents these transforms in rows (P ), (P ) and (P ) corresp ondingly . The \ hspacet op t∗{\ o bottomf i l l } gives$ C an impressive{\breve mixture{\sigma of}} many= common\ l e f t properties(e\ beginp { openarray parenthesish }{ cc } 1 e period & − g periode { the1 number}\\\ ofbreve fixed{\sigma} row (P ) e point{ 1 on} & thep boundary1 \end{ array for each}\ subgroupright ) closing , $ parenthesis\quad where with several\quad gradual$ \breve mutations{\sigma period} = − 1 , 0 , 1 $ almost coincides with Figs .1(A ), 1(N ) and 2(K ). Consideration of Fig . 1 7 by columns \quadThe descriptionand \quad ofdet the parabolic $ C {\ quotedblleftbrevea {\ unitsigmaa disk}} quotedblright=p 1 $ admits\quad severalf o r different a l l $ interpretations\breve{\sigma in t erms} { . } ( 8 . 9 )from $ lengthst op from t o bottomDefinition gives 5 period an 5impressive period mixture of many common properties ( e . g . the number Lemma 8 period 2 period .. Parabolic Cayley transform P sub sigma-breve as defined by the matrix C sub breve-sigma open parenthesis 8 \noindentof fixedHere $ \breve{\sigma} = − 1 $ corresp onds t o the parabolic Cayley transform $ P { e }$ periodpoint 9 closing on parenthesis the boundary acts on for the each V hyphen subgroup axis ) with several gradual mutations . withalways the as elliptic a shi f-t one flavour unit down $period , \breve{\sigma} = 1 $ −− t o theThe parabolic description Cayley of the transform parabolic “ $ unit P { diskh } ”$ admits with several the hyperbolic different interpretations flavour , cf in . t\quad [ 5 \quad 1 , \quad \S 2 . 6 ] . \quad F i n a l l y the Its imageerms can lengths be described from Definition in term of various 5 . 5 . lengths as fo l lows : p a1 r period a b o l i cP− subparabolic sigma-breve fortransform breve-sigma is equal-negationslash given by an upper 0 transforms− triangular the .. quotedblleft matrix real from axis the quotedblright end of the U to previous the p hyphen Lemma 8 . 2 . Parabolic Cayley transform P as defined by the matrix C (8.9) acts on cycleparagraph with the p . hyphen length squared minus sigma-breve from σ˘ σ˘ the V − axis always as a shi f − t one unit down . its e hyphen centreIts open image parenthesis can be described0 comma minus in term sigma-breve of various divided lengths by 2 as closing fo l parenthesis lows : comma .. cf period open parenthesis 8 period\ hspace 6 closing∗{\ f i parenthesis l l } Fig . :\quad 1 7 presents these transforms in rows $ ( P { e } ),(P { p } ) $ 1.P for σ˘ 6= 0 transforms the “ real axis ” U to the p - cycle with the p - length andEquation: $ ( open Pσ˘ { parenthesish } ) 8$ period corresp 10 closing ondingly parenthesis . \ ..quad openThe brace row open $parenthesis ( P u{ commap } v) closing $ parenthesis bar v = 0 closing squared −σ˘ from its e - centre (0, − σ˘ ), cf . ( 8 . 6 ) : brace right arrow braceleftbig open parenthesis u to2 the power of prime comma v to the power of prime closing parenthesis bar l sub c sub e\noindent to the poweralmost of 2 parenleftbig coincides parenleftbig with Figs 0 comma $. minus 1 sigma-breve ( A divided{ a } by) 2 parenrightbig , 1 ( comma N open{ a parenthesis} ) $ and u to the $ power 2 ( Kof prime{ p comma} ) v to . $the\ powerh f i l ofl primeConsideration closing parenthesis of Fig parenrightbig . \ h f iσ˘ l l times1 7 open\ h f i parenthesis l l by columns minus breve-sigma from closing parenthesis = 1 {(u, v) | v = 0} → {(u0, v0) | l2 ((0, − ), (u0, v0)) · (−σ˘) = 1}, (8.10) bracerightbig comma ce 2 \noindentwhere l subt c op sub t e to o the bottom power givesof 2 open an parenthesis impressive open parenthesis mixture of 0 comma many minus common sigma-breve properties divided ( by e 2 . closing g . theparenthesis number comma of fixed l2 ((0, − σ˘ ), (u0, v0)) = u02 +σv ˘ 0, open parenthesiswhere c ue to the2 power of prime commas v ee to the( 5 power . 5 ) . of prime Theclosing image parenthesis of upper halfplane closing parenthesis is : = u to the power of prime 2 plus\noindent breve-sigmapoint v to onthe powerthe boundary of prime comma for eachs ee open subgroup parenthesis ) with5 period several 5 closing gradual parenthesis mutations period . The image of upper halfplane is : σ˘ TheEquation: description open parenthesis of the{(u, v 8parabolic) period| v 1 > 10 closing} ‘‘ → unit {(u parenthesis0, v0 disk) | ’’l ..2 ((0 open admits, − brace), ( severalu0 open, v0)) parenthesis· (− differentσ˘) < 1}. u comma interpretations v closing(8.11) parenthesis in bart erms v greater 0 ce 2 closinglengths brace from right arrowDefinition braceleftbig 5 . open 5 . parenthesis u to the power of prime comma v to the power of prime closing parenthesis bar l sub c sub e to2 the.P powerσ˘ with of 2σ˘ parenleftbig6= 0 transforms parenleftbig the 0 “ comma real axis minus ” sigma-breveU to the p divided - cycle by with 2 parenrightbig p - length squared comma open parenthesis u to the \noindent Lemma 8 . 2 . \quad Parabolicσ˘ Cayley transform $ P {\breve{\sigma}}$ as defined by the matrix power of−σ˘ prime(5.7) from comma its vto h -the focus power(0 of, prime−1 − closing4 ), and parenthesis the upper parenrightbig half - plane times – to open the parenthesis “ interior minus ” breve-sigma part closing parenthesis less$ C 1 bracerightbig{\of itbreve , cf .{\ periodsigma( 8 . 6}} ) : ( 8 . 9 )$ actsonthe $V − $ a x i s always2 period as P sub a shi sigma-breve $ f−t with $ breve-sigma one unit equal-negationslash down . 0 transforms the .. quotedblleft real axis quotedblright U to the p hyphen cycle with p hyphen length squared minus sigma-breve open parenthesis 5 period 7 closing parenthesis \ centerlinefrom its h hyphen{ Its imagefocus open can parenthesis be described 0 comma in minus term 1 minus of various sigma-breveσ˘ ) lengths divided as by 4 fo closing l lows parenthesis : } comma and the upper half {(u, v) | v = 0} → {(u0, v0) | l2 ((0, −1 − , (u0, v0)) · (−σ˘) = 1}, (8.12) hyphen plane endash to the .. quotedblleft interior quotedblrightfh part of it4 comma cf period open parenthesis 8 period 6 closing parenthesis : $Equation: 1 . open P {\ parenthesisbreve{\ 8sigma period}} 1 2$ closing f o r parenthesis $ \breve .. open{\sigma braceσ˘}\ openne parenthesis0 $ u transforms comma v closing the parenthesis\quad ‘‘ bar real v = 0 closingaxis ’’ $ U $ to the p − cycle{(u, v) with| v >the0} →p − {(ulength0, v0) | squaredl2 ((0, −1 − $ −), (u0 \,breve v0)) · (−{\σ˘)sigma< 1}.} $ from(8.13) brace right arrow braceleftbig open parenthesis u to the powerf ofh prime comma4 v to the power of prime closing parenthesis bar l sub f h to the poweri t s of e − 2 parenleftbigc e n t r e $ parenleftbig ( 0 0 , comma− minus\ f r a c 1{\ minusbreve sigma-breve{\sigma}}{ divided2 by} 4 to) the , power $ \ ofquad parenrightbigc f . ( 8comma . 6 open ) : parenthesis u to the power of prime comma v to the power of prime closing parenthesis parenrightbig times open parenthesis minus breve-sigma closing parenthesis\ begin { a l = i g n1∗} bracerightbig comma Equation: open parenthesis 8 period 13 closing parenthesis .. open brace open parenthesis u comma v\{ closing( parenthesis u , bar v v greater ) \ 0mid closingv brace = right 0 arrow\}\ braceleftbigrightarrow open parenthesis\{ u( to the u ˆ power{\prime of prime} comma, v v ˆ to{\ theprime power} of) prime\mid closingl parenthesis ˆ{ 2 } { barc l{ sube f}} sub h( to the ( power 0 of , 2 parenleftbig− \ f r a c {\ parenleftbigbreve{\ 0sigma comma}}{ minus2 } 1 minus) sigma-breve , ( u divided ˆ{\prime by 4 } parenrightbig, v ˆ{\prime comma} open)) parenthesis\cdot u to the( power− of prime \breve comma{\sigma v to} the power) = of prime 1 closing\} parenthesis, \ tag ∗{$ parenrightbig ( 8 . times 10 open ) $} parenthesis\end{ a l i g minusn ∗} breve-sigma closing parenthesis less 1 bracerightbig period \noindent where $ l ˆ{ 2 } { c { e }} ( ( 0 , − \ f r a c {\breve{\sigma}}{ 2 } ) , ( u ˆ{\prime } , v ˆ{\prime } ) ) = u ˆ{\prime 2 } + \breve{\sigma} v ˆ{\prime } ,$ see(5.5). The image of upper halfplane is :

\ begin { a l i g n ∗} \{ ( u , v ) \mid v > 0 \}\rightarrow \{ ( u ˆ{\prime } , v ˆ{\prime } ) \mid l ˆ{ 2 } { c { e }} ( ( 0 , − \ f r a c {\breve{\sigma}}{ 2 } ) , ( u ˆ{\prime } , v ˆ{\prime } )) \cdot ( − \breve{\sigma} ) < 1 \} . \ tag ∗{$ ( 8 . 1 1 ) $} \end{ a l i g n ∗}

$ 2 . P {\breve{\sigma}}$ with $ \breve{\sigma}\ne 0 $ transforms the \quad ‘‘ real axis ’’ $ U $ to the p − c y c l e with p − length squared $ − \breve{\sigma} ( 5 . 7 ) $ from i t s h − f o c u s $ ( 0 , − 1 − \ f r a c {\breve{\sigma}}{ 4 } ) , $ and the upper half − plane −− to the \quad ‘‘ interior ’’ part of it , cf . ( 8 . 6 ) :

\ begin { a l i g n ∗} \{ ( u , v ) \mid v = 0 \}\rightarrow \{ ( u ˆ{\prime } , v ˆ{\prime } ) \mid l ˆ{ 2 } { f h } ( ( 0 , − 1 − \ f r a c {\breve{\sigma}}{ 4 }ˆ{ ) } , ( u ˆ{\prime } , v ˆ{\prime } )) \cdot ( − \breve{\sigma} ) = 1 \} , \ tag ∗{$ ( 8 . 1 2 ) $}\\\{ ( u , v ) \mid v > 0 \}\rightarrow \{ ( u ˆ{\prime } , v ˆ{\prime } ) \mid l ˆ{ 2 } { f { h }} ( ( 0 , − 1 − \ f r a c {\breve{\sigma}}{ 4 } ) , ( u ˆ{\prime } , v ˆ{\prime } )) \cdot ( − \breve{\sigma} ) < 1 \} . \ tag ∗{$ ( 8 . 13 ) $} \end{ a l i g n ∗} Erlangen Program at Large hyphen 1 : .... Geometry of Invariants .... 4 1 \noindenthlineErlangenErlangen Program Program at Large at - Large 1 :− 1 : \ h f i l l GeometryGeometry of of Invariants Invariants \ h f i l l 4 1 4 1 3 period P sub sigma-breve transforms the .. quotedblleft real axis quotedblright U to the cycle with p hyphen length minus breve-sigma from\ [ \ itsr u pl e hyphen{3em}{ focus0.4 pt open}\ parenthesis] 0 comma minus 1 closing parenthesis comma and the upper half hyphen p lane endash to the .. quotedblleft interior quotedblright part of it comma .. cf period open parenthesis 8 period 6 closing parenthesis3.Pσ˘ transforms : the “ real axis ” U to the cycle with p - length −σ˘ from its p - focus $Equation: 3(0, − .1), open Pand{\ parenthesis thebreve upper{\ 8sigma period half -}} 14 p laneclosing$ transforms – parenthesis to the .. the“ open interior\quad brace ” open‘‘ part real parenthesis of it axis , u cf’’ comma . $U$( 8 v . closing 6) to: parenthesis the cycle bar with v = 0 p closing− length brace$ − right \breve arrow{\ braceleftbigsigma} $ open from parenthesis i t s p u− tof o the c u spower $ of ( prime 0 comma , − v to the1 power ) of , prime $ and closing parenthesis bar l sub f sub p to thethe power upper of 2 open half parenthesis− p lane open−− parenthesisto the \ 0quad comma‘‘ minus interior 1 closing ’’ parenthesis part of comma it , open\quad parenthesisc f . ( u 8 to . the 6 power ) : of prime comma {(u, v) | v = 0} → {(u0, v0) | l2 ((0, −1), (u0, v0)) · (−σ˘) = 1}, (8.14) v to the power of prime closing parenthesis closing parenthesis timesfp open parenthesis minus sigma-breve closing parenthesis = 1 bracerightbig \ begin { a l i g n ∗} 0 0 2 0 0 comma Equation: open parenthesis{(u, v) 8| periodv > 0 1} 5 → closing {(u , v parenthesis) | lf ((0 .. open, −1), brace(u , v open)) · (− parenthesisσ˘) < 1}, u comma v closing(8.15) parenthesis bar v greater \{ ( u , v ) \mid v = 0 \}\p rightarrow \{ ( u ˆ{\prime } , v ˆ{\prime } 0 closing brace right arrow braceleftbig open parenthesis u to the power of prime comma02 v to the power of prime closing parenthesis bar l sub ) \mid l ˆ{ 2 } { f { p }} ( ( 02 , − 10 0 )u , ( u ˆ{\prime } , v ˆ{\prime } ) f sub p to the power of 2 open parenthesis open parenthesiswherelf 0((0 comma, −1), minus(u , v )) 1 = closing parenthesis(5.7). comma open parenthesis u to the power of p v0 + 1 prime) \ commacdot v to( the− power \breve of prime{\sigma closing} parenthesis) = closing 1 parenthesis\} , \ timestag ∗{ open$ ( parenthesis 8 . minus 14 sigma-breve ) $}\\\{ closing parenthesis( u , less 1v bracerightbig )Remark\mid comma 8v . 3 where> . l0 subNote f\}\ sub that p torightarrow the the both power elliptic of 2 open\{ ( 8 parenthesis .( 6 ) u and ˆ{\ open hyperbolicprime parenthesis} ( 8, 0 . comma 8 v ) ˆ unit{\ minusprime circle 1 closing} ) parenthesis\mid commal ˆ{ 2 } { f { p }} open( parenthesis(descriptions 0 , u to can− the power b1 e written )of prime , uniformly comma ( u v ˆto with{\ theprime power parabolic} of prime, descriptions closing v ˆ{\ parenthesisprime ( 8 .} 1 0 closing)) ) – ( parenthesis 8 . 1\cdot 4 ) as = u( to the− power \breve of prime{\sigma 2 } divided) < by1 v to the\} power, \ oftag prime∗{$ plus ( 1 open8 parenthesis . 1 5 5 period ) $}\\ 7 closingwhere parenthesis l ˆ{ period2 } { f { p }} ( ( 0 , − 1 ) , ( u ˆ{\prime } , v0 ˆ{\0 prime2 }0 )0 ) = \ f r a c { u ˆ{\prime 2 }}{ v ˆ{\prime } + 1 } Remark 8 period 3 period .. Note that{(u the, v both) | ellipticlcσ˘ (u opene0 + v parenthesise1) · (−σ˘) = 8 period1}. 6 closing parenthesis and hyperbolic open parenthesis 8 period( 5 8 closing . parenthesis7 ) . unit circle descriptions can \endb e{ a written l iThe g n ∗} uniformlyabove descriptions with parabolic 8 . descriptions2 . 1 and 8 open . 2 . parenthesis 3 are attractive 8 period for 1 reasons 0 closing given parenthesis in the endash following open parenthesis 8 period 1 4 closingtwo parenthesis lemmas as . Firstly , the K− orbits in the elliptic case ( Fig .18(Ke)) and the A− orbits in \noindentbraceleftbigthe hyperbolicRemark open parenthesis 8 case . 3 ( . Fig u\quad to. the18(ANote powerh)) of thatof Cayley prime the comma transform both v ellipticto arethe powerconcentric ( of 8 prime . 6. )closing and parenthesis hyperbolic bar ( l sub 8 . c 8 sigma-breve ) unit circle to the descriptions can powerb e of written 2 open parenthesis uniformly u to with the power parabolic of prime descriptions e sub 0 plus v to ( the 8 . power 1 0 of ) prime−− ( e 8 sub . 1 1 closing 4 ) as parenthesis times open parenthesis minus breve-sigma closing parenthesis = 1 bracerightbig sub period 1 concentric \ [ \{ ( u ˆ{\prime } −,orbits v ˆin{\primethe } cases) (F\ igmid l ˆ. {1 (2N P}e, { Pcp \breve{\sigma}} ( u ˆ{\prime } TheLemma above descriptionslas(or8.4. Nstraight 8 period 2lines period) in 1 andthe 8 periodparabolic 2 periodsense 3ofDefinition are attractive2.172 forwith reasonsNe−centres given,N inPh the)@) followingare(0, two parabo(0,− 21−) 2 ), (0,∞), e lemmas{ 0 } period+ Firstlyv ˆ{\ commaprime the} Ke hyphen{ 1 } orbits) in\ thecdot elliptic( case− open \ parenthesisbreve{\sigma Fig period} ) 1 8 open = parenthesis 1 \} { K. sub}\ e] closing parenthesiscorrespondingly closing parenthesis . and the A hyphen orbits in the hyperbolic case openSecondly parenthesis , Fig Calley period images 1 8 open parenthesis of the Afix sub hsubgroups closing parenthesis ’ orbits closing in parenthesis elliptic of Cayley and transform are concentric periodThe above descriptions 8 . 2 . 1 and 8 . 2 . 3 are attractive for reasons given in the following two lemmashyperbolic . Firstly spaces , the in$ K − $ orbits in the elliptic case (Fig $ . 1 8 ( K { e } ) ) $ LemmaFig las.18( openAh) parenthesisand (Ke) are or 8 equidistant period 4 period from N straight the origin sub linesin the closing corresponding parenthesis to metrics the power . of hyphen orbits in to the power of inand the the to the $ power A of− the$ parabolic orbits s in ense the of Definition hyperbolic to the power of cases open parenthesis Fig sub 2 period to the power of period sub 1 case(FigLemma 8$. . 5 . 1 8The ( A Cayley{ h } transform) ) $ of orbits of Cayley of the transform parabolic fix are subgroup concentric in Fig . to the power of0 1 7 2 to the power of open parenthesis N sub with to the power of P sub e comma N e hyphen centres to the power of P sub p . 18(NP ) are comma N sub Pe sub h closing parenthesis at closing parenthesis are open parenthesis sub 0 comma 1 divided by 2 to the power of concentric sub \ [Lemmaparabolas{ l a s } consisting( or of points 8 . on 4the same . N l{fp −s tlength r a i g h t (} 5ˆ{ . 6 − ) fromo r b the i t s point} { l i(0 n e, − s1), ) cf} . in ˆ{ in } the ˆ{ the } closing8 parenthesis . 2 . 3 . comma open parenthesis 0 comma infinity closing parenthesis comma parabo open parenthesis sub 0 comma minus overbar 2p 1a r hyphen a b o l i c closing{ s parenthesis ense of De fi nition }ˆ{ c a s e s ( Fig }ˆ{ . } { 2 . }ˆ{ 1 } { 1 } 7 { 2 }ˆ{ ( N }ˆ{ P { e } Note, } that{ with parabolic} N rotations{ e − of thec e n parabolict r e s }ˆ{ unitP { diskp }} are incompatible,N { P with{ h }} the ) { @ } ) are { ( } { 0 correspondinglyalgebraic period , }\Secondlyf r a c { comma1 }{ ..2 Calley}ˆ{ imagesc o n c e n.. t of r i cthe} ..{ fix) .. subgroups , ( quoteright 0 , ..\ orbitsinfty .. in ..), elliptic} ..parabo and .. hyperbolic{ ( } { ..0 spaces , ..− in \ overline {\}{ 2 }} 1 −{structure) }\ ] provided by the algebra of dual numbers . However we can introduce [ 4 8 , 4 7 Fig] period a linear 1 8 open algebra parenthesis structure A sub and hclosing vector parenthesis multiplication and open which parenthesis will rotationally K sub e closing invariant parenthesis under are equidistant from the origin in the correspondingaction of metrics period Lemma 8 period 5 period ....0 The .... Cayley transform of orbits of the parabolic fix subgroup in Fig period 1 8 open parenthesis N sub P \noindentsubgroupscorrespondinglyN and N . . sub e toRemark the power 8 of . prime 6 . closingWe parenthesis see that the .... are varieties of possible Cayley transforms in the parabolic parabolas consisting of points on the same l sub f sub p hyphen length open parenthesis 5 period 6 closing parenthesis from the point open \ hspacecase∗{\ i sf i biggerl l } Secondly than in , the\quad two otherCalley cases images . It\quad is interestingo f the \ thatquad thisf i x parabolic\quad subgroups richness i s ’ \quad o r b i t s \quad in \quad e l l i p t i c \quad and \quad h y p e r b o l i c \quad spaces \quad in parenthesisa consequence 0 comma minus 1 closing parenthesis comma .. cf period 8 period 2 period 3 period Note that parabolic rotations of the parabolic unit disk are incompatible with the algebraic \noindentof theFig parabolic $ . degeneracy 1 8 ( of A the{ generatorh } ) $2e1 and= $ 0. (Indeed K { fore both} ) the $ are elliptic equidistant and from the origin in the corresponding metrics . structurethe provided by the algebra of dual numbers period .. However we can introduce open square bracket 4 8 comma .. 4 7 closing square   bracket a linear 1 e1 \noindenthyperbolicLemma signs 8 . in 52 . e1\=h± f i1 l lonlyThe one\ h matrix f i l l Cayley ( 8 . 1 transform ) out of two of possible orbits of the parabolichas a fix subgroup in Fig algebra structure and vector multiplication which will rotationally invariant .. under action of±σe1 1 $ .subgroupsnon 1 - 8 N and ( N to N the ˆ{\ powerprime of prime} { periodP { e }} ) $ \ h f i l l are

Remarkzero 8 determinant period 6 period . .. We And see only that for the the varieties degenerate of possible parabolic Cayley transforms value 2e1 in= the 0 both parabolic these case matrices i s \noindentbiggerare than nonparabolas in - the singular two other consisting ! cases period of .. points It is interesting on the that same this parabolic $ l { richnessf { p i s}} a consequence − $ length ( 5 . 6 ) from the point $ (of the8 0 . parabolic 3 , Cayley− .... degeneracy1 transforms ) .... , $ of the\ ofquad generator cyclesc f . .... 8 2 . e sub 2 . 1 3= 0 . period .... Indeed for both the .... elliptic .... and .... the hyperbolicThe next signs natural in 2 e sub step 1 = within plusminux the 1 FSCc only one i s matrix t o expand open parenthesis the Cayley 8 period transform 1 closing t parenthesis o the space out of of two possible .... Row 1 1 e\ hspace sub 1cycles Row∗{\ 2f . plusminux i l This l }Note is sigma performed that e sub parabolic 1 as 1 .follows .... has rotations : a non hyphen of the parabolic unit disk are incompatible with the algebraic zero determinant period .. And only for the degenerate parabolicσ value 2 e sub 1 = 0 both these matrices are \noindentLemmastructure 8 . 7 . providedLet C bys be the a cycle algebra in R of dual numbers . \quad However we can introduce [ 4 8 , \quad 4 7 ] a l i n e a r non hyphen singular ! a . algebra structure and vector multiplication which will rotationally invariant \quad under action of 8 period 3 .. Cayley transforms of cycles s ( e , h ) In the e l lip tic or hyperbolic cases the Cayley transform maps a cycle Cσ˘ to the Thecomposition next natural step within the FSCc i s t o expand the Cayley transform t o the space of cycles period \noindentThis is performedsubgroups as follows $N$ : and $Nˆ{\prime } . $   ˆs sCˆ s s ˆs ±e˘1 1 of its inversion with the reflection C C in the cycle C , where C = Lemma 8 period 7 period .. Let C sub a to the powerσ˘ ofσ˘ s beσ˘ a cycle in Case 1 sigmaσ˘ Case 2σ period˘ 1 ∓e˘1 \noindentopen parenthesisRemark e comma 8 . 6 h .closing\quad parenthesisWe see .... that In the the e l lip varieties tic or hyperbolic of possible cases the Cayley Cayley transform transforms maps a cycle in the C sub parabolic sigma-breve case i s biggerwith than in the two other cases . \quad It is interesting that this parabolic richness i s a consequence to the power of s to the compositionσ˘ = ±1( s ee the first and last drawings on Fig . 1 9 ) . of its inversion with the reflection C-circumflex sub breve-sigma to the power of s C sub sigma-breve to the power of s to the power of \noindent of the( p ) parabolic In the\ h f parabolic i l l degeneracy case\ h f the i l l of Cayley the generator transform\ h f i l l maps$ 2 { ae { 1 }} = 0 . $ circumflex-Ccycle sub sigma-breve(k, l, n, m) toto the power the of s cycle in the cycle(k C− sub breve-sigma to the power of comma to the power of s where C-circumflex sub breve-sigma\ h f i l l Indeed to thepower for both of s = the Row\ 1h plusminux f i l l e l l ie-breve p t i c sub\ h f 1 i 1l l Rowand 2 1\ minusplush f i l l the to the power of breve-e 1 . .. with sigma-breve = plusminux 1 open parenthesis s ee the first and last drawings on Fig period 1 9 closing parenthesis period \noindentopen parenthesishyperbolic p closing signs parenthesis in ..$ In 2 ..{ thee parabolic{ 1 }} .. case= ..\pm the .. Cayley1 $ ..only transform one matrix .. maps .. ( a 8 .. . cycle 1 ) .. out open of parenthesis two possible k \ h f i l l comma$\ l e f t l( comma\ begin n{ commaarray }{ mcc closing} 1 parenthesis & e { ..1 to}\\\ .. thepm .. cycle\ ..sigma open parenthesise { 1 k} minus& 1 \end{ array }\ right )$ \ h f i l l has a non − 2 sigma-breve n comma l comma n comma m plus 2 breve-sigma n closing parenthesis period \noindent zero determinant . \quad And only for the degenerate parabolic value $ 2 { e { 1 }} = 0 $ both these matrices are non − s i n g u l a r !

\noindent 8 . 3 \quad Cayley transforms of cycles

\noindent The next natural step within the FSCc i s t o expand the Cayley transform t o the space of cycles . This is performed as follows :

\noindent Lemma 8 . 7 . \quad Let $ C ˆ{ s } { a }$ bea cycle in $\ l e f t .R\ begin { a l i g n e d } & \sigma \\ &. \end{ a l i g n e d }\ right . $

\noindent ( e , h ) \ h f i l l In the e l lip tic or hyperbolic cases the Cayley transform maps a cycle $ C ˆ{ s } {\breve{\sigma}}$ to the composition

\ hspace ∗{\ f i l l } of its inversion with the reflection $ \hat{C} ˆ{ s } {\breve{\sigma}} C ˆ{ s } {\breve{\sigma}}ˆ{\hat{C}}ˆ{ s } {\breve{\sigma}}$ in the cycle $Cˆ{ s } {\breve{\sigma} ˆ{ , }}$ where $ \hat{C} ˆ{ s } {\breve{\sigma}} = \ l e f t (\ begin { array }{ cc }\pm \breve{e} { 1 } & 1 \\ 1 & \mp ˆ{\breve{e}} 1 \end{ array }\ right )$ \quad with

\ centerline { $ \breve{\sigma} = \pm 1 ( $ s ee the first and last drawings on Fig . 1 9 ) . }

\ hspace ∗{\ f i l l }( p ) \quad In \quad the parabolic \quad case \quad the \quad Cayley \quad transform \quad maps \quad a \quad c y c l e \quad $(k,l ,n,m)$ \quad to \quad the \quad c y c l e \quad $ ( k − $

\ [ 2 \breve{\sigma} n , l , n , m + 2 \breve{\sigma} n ) . \ ] 2˘σn, l, n, m + 2˘σn). 42 .... V period V period Kisil \noindenthline42 42 \ h f i l l V . V . K i s i l V . V . Kisil Figure 19 period .. Cayley transforms in elliptic open parenthesis sigma = minus 1 closing parenthesis comma parabolic open parenthesis sigma\ [ \ r = u l 0 e closing{3em}{ parenthesis0.4 pt }\ and] hyperbolic open parenthesis sigma = 1 closing parenthesis spaces period On the each picture the reflection of the real line in the green cycles open parenthesis drawn continuously or dotted closing parenthesis is theFigure is the blue 19 . .. quotedblleftCayley transforms unit cycle in quotedblright elliptic (σ = − period1), parabolic Reflections (σ = in 0) the and solidly hyperbolic drawn (σ cycles= 1) spaces send the . On upper half hyphen plane to the\noindent unitthe eachFigure picture 19the reflection . \quad ofCayley the real transformsline in the green in cycles elliptic ( drawn continuously $ ( \sigma or dotted= ) is− the is1 the ) ,$ parabolic $ (diskblue comma\sigma “ unitreflection= cycle in0 ” . the Reflections )$ dashed and cycle in hyperbolicthe endash solidly to drawn its complement $( cycles\ sendsigma period the upper Three= half Cayley 1 - plane )$ transforms to spaces. the unit in the parabolic space Onopen thedisk parenthesis each , reflection picture sigma in the = the 0 dashed closing reflection cycle parenthesis – to its of are complement the themselves real . line elliptic Three in openCayley the parenthesis greentransforms cycles sigma-breve in the ( parabolic drawn = minus spacecontinuously 1 closing parenthesis or dotted comma ) is parabolicthe( isσ = open the 0) are parenthesis blue themselves\quad breve-sigma elliptic‘‘ unit (˘σ = cycle− 01) closing, parabolic ’’ parenthesis . Reflections (˘σ = 0) and and hyperbolic hyperbolic in the open (˘σ solidly= parenthesis 1), giving drawn a sigma-breve gradual cycles = send 1 closing the parenthesis upper half comma− plane to the unit giving atransition gradual between proper elliptic and hyperbolic cases . \noindenttransitionThe betweendisk above , proper reflection extensions elliptic and in of hyperbolicthe the Cayley dashed cases transform cycle period−− t oto the its cycles complement space i s .linear Three , however Cayley in transforms in the parabolic space $The (the above\sigma extensions= of the 0 Cayley ) $ transform are themselves t o the cycles elliptic space i s linear $ ( comma\breve however{\sigma in the} = − 1 ) ,$ parabolic $ (parabolicparabolic\breve case{\ it casesigma i s not it} i expressed s not= expressed 0 as a similarity)$ as and a ofsimilarity hyperbolic matrices open of matrices parenthesis $( (\ reflectionsbreve reflections{\sigma in in a a cycle} cycle= closing ) . 1 parenthesis This ) ,$ period givingagradual .... This can b ecan seen comma for example comma from the fact that the parabolic Cayley transform does not preserve the \noindentzerob hyphen e seentransition radius , for cycles example between represented , from proper by the matrices fact elliptic that with the zero and parabolic p hyphen hyperbolic determinant Cayley cases transform period . does not preserve Sincethe .. orbits zero - .. radius of all subgroups cycles represented in SL sub 2 open by matrices parenthesis with R closing zero p parenthesis - determinant .. as well . as their Cayley images .. are .. cycles in the \ hspacecorresponding∗{\ f i l l metricsSince}The above we orbits may extensions use ofLemma all subgroups 8 of period the 7 Cayley in openSL2 parenthesis(R transform) as well p closing tas o their the parenthesis Cayley cycles timages spaceo prove i the are s following linear statements , however open in the parenthesiscycles in inaddition the \noindentt ocorresponding Lemmaparabolic 8 period 4metrics closing case parenthesiswe it may i s use not : Lemma expressed 8 . 7 as ( p a ) similarity t o prove the of following matrices statements ( reflections ( in in a cycle ) . \ h f i l l This can Corollaryaddition 8 period 8 period \noindentEquation:t o Lemmab A hyphene seen 8 . orbits 4 , ) :for through example sub open , from parenthesis the sub fact 0 comma that to the the parabolic power of in transforms Cayley transformminus 1 divided does by 2 not closing preserve parenthesis the periodzeroCorollary Their− radius to the power 8 cycles . 8 of . P represented sub vertices to by the matricespower of e to with the power zero of p and− determinant P sub belong to . the power of h to the power of are s sub to to the power of egments 2 parabolas to the power of of parabolas v with = 1 divided by 2 sub open parenthesis minus x to the power of the 2 focal\ hspace minus∗{\ 1 closingf i l l } Since parenthesis\quad ando to r b the i t s power\quad of lengthof all v = subgroups 2 from 1 to 1 in divided $ SL by 2 sub{ 2 open} parenthesis( R ) to $ the\ powerquad ofas hline well sub asx to their Cayley images \quad are \quad cycles in the 1. 1 in the powerP of 2 toe the powerandPh of passingaresegments sub minus 1 closingofparabolas parenthesis .. 11 the period length 21 passing A − orbitsthrough( 0,transforms\noindent− 1 ). T heircorrespondingvertices belong metricsto we2parabolas may use Lemmavwith 8 .= 72 ( (−x p2focal ) t−1) o proveandv = the2 ( followingx2 −1) statements ( in addition correspondingly2 comma which are boundaries of parabolic circles in P sub h .. and P sub e open parenthesis note the swap ! closing parenthesis period \noindentcorrespondingly2 period Kt hyphen o Lemma , orbits which 8 in . are transform 4 boundaries ) : P sub ofe are parabolic parabolas circles with focal in lengthPh and less thanPe( 1note divided the by swap 2 and ! in) transform. P sub h endash 2.K− P 1 with inverse of focal lengthorbits bigger in than transform minus 2 periode are parabolas with focal length less than 2 and in \noindentSincetransform the actionCorollary ofPh parabolic– 8 . Cayley 8 . transform on cycles does not preserve zero hyphen radius cyc hyphen les one shall b etter use infinitesimalwith hyphen inverse radius of focal cycles length from Section bigger ..than 6 period−2. 1 instead period .. First of all ima hyphen \ beginges of{ infinitesimala l i gSince n ∗} the cycles action under of parabolic parabolic Cayley Cayley transform transform are infinitesimal on cycles cyclesdoes not again preserve open square zero bracket - radius 4 6 comma 1S 3cyc . period\ tag - 6∗{ period$ A 4− closingo squarer b i t s bracket{ through comma} ..{ secondly( }ˆ{ Lemmain } { ..0 6 period , } 5 periodtransforms 2 provides{ a − useful \ f r expression a c { 1 }{ of2 concurrence} ). withTheir infinitesimalles}ˆ one{ P shall}ˆ{ be etter} { usev e r infinitesimal t i c e s }ˆ{ and - radius P cycles}ˆ{ h from} { Sectionbelong }ˆ 6{ .are 1 instead s } .ˆ{ egments First of all} { to } 2 parabolas ˆ{ o f parabolascycleima focus -} ges throughv of infinitesimal with f hyphen{ = orthogonality\ f cycles r a c { under1 }{ period2 parabolic}} .. Although{ ( Cayley− f hyphenx transform}ˆ orthogonality{ the are} infinitesimal2 i s not f o preserved c a l cycles{ − by again the1 Cayley ) and }ˆ{ length } vtransform =[ 4\ 6f r , a 8 c§ period3{ .2 6 ˆ .7{ open41 ] ,} parenthesis{ secondly1 }}{ p2 closingLemma}ˆ{\ parenthesisr u l 6e { .3em 5 . for}{ 20.4 generic provides pt }} cycles a{ useful it( did}ˆ{ for expressionpassing the infinitesimal of} concurrence{ x ones ˆ{ comma2 }} { see − open1 square ) bracket}$} 4\end 6 comma{witha l i g S n infinitesimal 3∗} period 6 period cycle 4 closing focus square through bracket f - orthogonality : . Although f - orthogonality i s not Lemmapreserved .... 8 period by the 9 period Cayley .... An transform infinitesimal 8 . cycle 7 ( C p sub ) for sigma-breve generic cycles to the power it did of for a open the parenthesis infinitesimal 6 period 1 closing parenthesis ....\ centerline is fones hyphen , see{ orthogonalcorrespondingly [ 4 6 , § open3 . 6 parenthesis . 4 ] , : which in the are s ense boundaries .... of Lemma of 6 period parabolic 5 period circles 2 closingin parenthesis $ P { h }$ \quad and $ P { e } ( $ note the swap ! ) . } a to ..Lemma a .. cycle C-tilde 8 . 9 sub . breve-sigmaAn infinitesimal to the power cycle of aC ..σ˘ if(6 and.1) onlyis f if- orthogonalthe .. Cayley( transformin the s 8 ense period 7 of open parenthesis p closing parenthesisLemma .. of C6 sub . 5 sigma-breve . 2 ) to the power of a .. is f hyphen orthogonal to .. the .. Cayley \ hspace ∗{\ f i l l } $ 2˜ .a K − $ orbits in transform $ P { e }$ are parabolasa with focal length less than transformto of a C-tilde cycle sub breve-sigmaCσ˘ if and to only the power if the of period Cayley to the transform power of a 8 . 7 ( p ) of Cσ˘ is f - $\ f r a c { 1 }{ 2 }$ and in transform $ P { h }$˜a −− Weorthogonal main observation to of the this paper Cayley is that the transform potential of of theCσ˘ Erlangen. programme i s still far fromWe exhausting main observation even for two of hyphen this paper dimensional is that geometry the potential period of the Erlangen programme i s still \ centerlineAcknowledgementsfar from{ with exhausting inverse even of for focal two - length dimensional bigger geometry than $ . − 2 . $ } ThisAcknowledgements paper has .. some .. overlaps with the paper .. open square bracket 5 1 closing square bracket .. written in .. collab oration with .. D period\ hspaceThis .. Biswas∗{\ paperf i lperiod l } Since has thesome action overlaps of parabolic with the paper Cayley transform [ 5 1 ] written on cycles in does collab not oration preserve zero − r a d i u s cyc − Howeverwith the D present . Biswas paper essentially . However revises the many present concepts paper .. open essentially parenthesis revises e period many g period concepts lengths comma ( e . .. orthogonality comma ..\noindent the g . lengthsles one ,orthogonality shall b etter , use the infinitesimal parabolic Cayley− radius transform cycles ) introduced from Section in [ 5 1\ ]quad , thus6 . 1 instead . \quad First of all ima − gesparabolicit of was infinitesimal Cayley important transform t o cycles closing make parenthesis it under an independent parabolic introduced Cayleyreadingin open square transformto avoid bracket confusion are5 1 closing infinitesimal with square some bracket earlier cycles comma ( againthus it was [ 4 important 6 , t o\ makeS 3and it . an 6 na independent .¨ 4ı ve ] ! , )\quad guessessecondly made Lemma in [ 5 1\ ]quad . 6 . 5 . 2 provides a useful expression of concurrence with infinitesimal cyclereading focus to avoid through confusion f with− orthogonality some earlier open parenthesis . \quad Although and na dieresis-dotlessi f − orthogonality ve ! closing parenthesis i s not .. preserved guesses made by in the open Cayley square brackettransform 5 1 closing 8 . square 7 ( p bracket ) for period generic cycles it did for the infinitesimal ones , see [ 4 6 , \S 3 . 6 . 4 ] : \noindent Lemma \ h f i l l 8 . 9 . \ h f i l l An infinitesimal cycle $ C ˆ{ a } {\breve{\sigma}} ( 6 . 1 ) $ \ h f i l l i s f − orthogonal ( in the s ense \ h f i l l o f Lemma 6 . 5 . 2 )

\noindent to \quad a \quad c y c l e $ \ tilde {C} ˆ{ a } {\breve{\sigma}}$ \quad if and only if the \quad Cayley transform 8 . 7 ( p ) \quad o f $ C ˆ{ a } {\breve{\sigma}}$ \quad i s f − orthogonal to \quad the \quad Cayley transform of $ \ tilde {C} ˆ{ a } {\breve{\sigma} ˆ{ . }}$

We main observation of this paper is that the potential of the Erlangen programme i s still far from exhausting even for two − dimensional geometry .

\noindent Acknowledgements

\noindent This paper has \quad some \quad overlaps with the paper \quad [ 5 1 ] \quad written in \quad collab oration with \quad D. \quad Biswas . However the present paper essentially revises many concepts \quad ( e . g . lengths , \quad orthogonality , \quad the parabolic Cayley transform ) introduced in [ 5 1 ] , thus it was important t o make it an independent reading to avoid confusion with some earlier ( and na $ \ddot{\imath} $ ve ! ) \quad guesses made in [ 5 1 ] . Erlangen Program at Large - 1 : Geometry of Invariants 43

The author is grateful t o Professors S . Plaksa , S . Blyumin and N . Gromov for useful discus - sions and comments . Drs . I . R . Porteous , D . L . Selinger and J . Selig carefully read the previous paper [ 5 1 ] and made numerous comments and remarks helping to improve this paper as well . I am also grateful to D . Biswas for many comments on this paper . The extensive graphics in this paper were produced with the help of the G i NaC [ 4 , 4 4 ] com - puter algebra system . Since this tool is of separate interest we explain it s usage by examples from this article in the separate paper [ 4 6 ] . The noweb [ 6 4 ] wrapper for C ++ source code i s included in the arXiv . org files of the papers [ 4 6 ]. References [ 1 ] Arveson W . , An invitation to C∗− algebras , Graduate Texts in Mathematics , Vol . 39 , Springer - Verlag , New York – Heidelberg , 1 976 . [ 2 ] Baird P . , Wood J . C . , Harmonic morphisms from Minkowski space and hyperbolic numbers , Bull . Math . Soc . Sci . Math . Roumanie ( N . S . ) 52 ( 100 ) ( 2009 ) , 1 9 5 – 209 . [ 3 ] Barker W . , Howe R . , Continuous symmetry . From Euclid to Klein , American Mathematical Society , Provi - dence , RI , 2 7 . [ 4 ] Bauer C . , Frink A . , Kreckel R . , Vollinga J , G i N a C is Not a CAS , h ttp : / / w w w . g in ac . de / . [ 5 ] Beardon A . F . , The geometry of discrete groups , Graduate Texts in Mathematics , Vol . 9 1 , Springer - Verlag , New York , 1 995 . [ 6 ] Beardon A . F . , Algebra and geometry , Cambridge University Press , Cambridge , 2005 . [ 7 ] Bekkara E . , Frances C . , Zeghib A . , On lightlike geometry : isometric actions , and rigidity aspects , C.R. M a t h . A cad . S ci . P a r i s 343 ( 2006 ) , 317 – 32 1 . [ 8 ] Benz W . , Classical geometries in modern contexts . Geometry of real inner product spaces , 2 nd ed . , Birkh a¨ user Verlag , Basel , 2 7 . [ 9 ] Berger M . , Geometry . II , Springer - Verlag , Berlin , 1 987 . [ 1 0 ] Boccaletti D . , Catoni F . , Cannata R . , Catoni V . , Nichelatti E . , Zampetti P . , The mathematics of Minkowski space - time and an introduction to commutative hypercomplex numbers , Frontiers in Mathe - matics , Birkh a¨ user Verlag , Basel , 2008 . [ 1 1 ] Catoni F . , Cannata R . , Nichelatti E . , The parabolic analytic functions and the derivative of real functions , A d v . A p pl . C liff o r d A lg e b r . 14 ( 2 4 ) , 1 85 – 1 90 . [ 1 2 ] Catoni F . , Cannata R . , Catoni V . , Zampetti P ., N− dimensional geometries generated by hypercomplex numbers , A dv . A ppl . C liff or d A lg e b r . 15 ( 2005 ) , 1 – 25 . [ 1 3 ] Cerej eiras P . , K a¨ hler U . , Sommen F . , Parabolic Dirac operators and the Navier – Stokes equations over time - varying domains , M a th . M e t h o d s A ppl . Sc i . 28 ( 2005 ) , 1 71 5 – 1 724 . [ 14 ] Chern S . - S . , Finsler geometry is j ust Riemannian geometry without the quadratic restriction , Notices Amer . Math . Soc . 43 ( 1 996 ) , 959 – 963 . [ 1 5 ] Cnops J . , Hurwitz pairs and applications of M o¨ bius transformations , Habilitation Dissertation , University of Gent , 1 994 . [ 1 6 ] Cnops J . , An introduction to Dirac operators on manifolds , Progress in Mathematical Physics , Vol . 24 , Birkh a¨ user Boston , Inc . , Boston , MA , 2002 . [ 1 7 ] Coxeter H . S . M . , Greitzer S . L . , Geometry revisited , Random House , New York , 1 9 67 . [ 1 8 ] Davis M . , Applied nonstandard analysis , John Wiley & Sons , New York , 1 977 . [ 1 9 ] Eelbode D . , Sommen F . , Taylor series on the hyperbolic unit ball , Bull . Belg . Math . Soc . Simon Stevin 1 1 ( 2 4 ) , 71 9 – 737 . [ 20 ] Eelbode D . , Sommen F . , The fundamental solution of the hyperbolic Dirac operator on R1,m : a new ap - proach , Bull . Belg . Math . Soc . Simon Stevin 12 ( 2005 ) , 23 – 37 . Erlangen Program at Large hyphen 1 : .... Geometry of Invariants .... 43 \noindent Erlangen Program at Large − 1 : \ h f i l l Geometry of Invariants \ h f i l l 43 hline [ 2 1 ] Fjelstad P . , Gal S . G . , Two - dimensional geometries , topologies , trigonometries and physics Thegenerated author is by grateful t o Professors S period Plaksa comma S period Blyumin and N period Gromov for useful discus hyphen \ [ \ r u l e {3em}{0.4 pt }\ ] sions and commentscomplex period - type ..numbers Drs period , Adv I. Applperiod . C R li periodff ord Porteous A lge comma br . 1 D 1 period( 2001 L ) period , 81 – 1 Selinger 7 . and J period Selig carefully read the previous[ 22 ] Garas ’ ko G . I . , Elements of Finsler geometry for physicists , TETRU , Moscow , 2009 ( in Russian ) , paperavailable open square bracket 5 1 closing square bracket .. and made numerous comments and remarks helping to improve this paper as well The author is grateful t o Professors S . Plaksa , S . Blyumin and N . Gromov for useful discus − at http: //hypercompperiod lex. xpsweb. com/articl es/ 487/ru/pdf/0-gbook. p df . sionsI am also and grateful comments to D period . \quad BiswasDrs for . many I . comments R . Porteous on this paper , D . period L . Selinger and J . Selig carefully read the previous paperThe extensive [ 5 1 graphics ] \quad inand this paper made were numerous produced comments with the help and of remarks the G i NaC helping open square to bracket improve 4 comma this .. paper 4 4 closing as well square . bracket comI am hyphen also grateful to D . Biswas for many comments on this paper . puter algebra system period .. Since this tool is of separate interest we explain it s usage by examples Thefrom extensive this article graphics in the separate in thispaper ..paper open square were producedbracket 4 6 closing with the square help bracket of period the G .. i The NaC noweb [ 4 .. , open\quad square44 bracket ] com 6 ..− 4 closingputer square algebra bracket system .. wrapper . \ forquad C plusSince plus sourcethis toolcode i sis of separate interest we explain it s usage by examples fromincluded this in the article arXiv period in the org separate files of the papers paper open\quad square[ 4 bracket 6 ] .4 ..\quad 6 closingThe square noweb bracket\quad period[ 6 \quad 4 ] \quad wrapper for C $+References +$ source code i s includedopen square in bracket the arXiv 1 closing .square org files bracket of .. theArveson papers W period [ 4 comma\quad An6 invitation] . to C to the power of * hyphen algebras comma Graduate Texts in Mathematics comma Vol period 39 comma Springer hyphen Verlag comma New \noindentYork endashReferences Heidelberg comma 1 976 period open square bracket 2 closing square bracket .. Baird P period comma Wood J period C period comma Harmonic morphisms from Minkowski space[ 1 ] and\quad hyperbolicArvesonW numbers . comma , An Bull invitation period Math to period $ C Soc ˆ{ period ∗ } − $ algebras , Graduate Texts in Mathematics , Vol . 39 , Springer − Verlag , New YorkSci period−− Heidelberg Math period Roumanie , 1 976 open . parenthesis N period S period closing parenthesis 52 open parenthesis 100 closing parenthesis open parenthesis 2009 closing parenthesis comma 1 9 5 endash 209 period \ hspaceopen square∗{\ f i bracket l l } [ 2 3 closing ] \quad squareBaird bracket P . .. ,Barker Wood W J period . C .comma , Harmonic Howe R period morphisms comma fromContinuous Minkowski symmetry space period and From hyperbolic Euclid numbers , Bull . Math . Soc . to Klein comma American Mathematical Society comma Provi hyphen \ centerlinedence comma{ Sci RI comma . Math 2 7 . period Roumanie (N . S . ) 52 ( 100 ) ( 2009 ) , 1 9 5 −− 209 . } open square bracket 4 closing square bracket .. Bauer C period comma Frink A period comma Kreckel R period comma Vollinga J comma G[ i 3 N ] a C\quad is NotBarker a CAS comma W . , .. Howe h ttp :R slash . , slash Continuous w w w period symmetry g in ac period . From de slash Euclid period to Klein , American Mathematical Society , Provi − denceopen square , RI bracket , 2 7 5 . closing square bracket .. Beardon A period F period comma The geometry of discrete groups comma Graduate Texts in Mathematics comma Vol period 9 1 comma Springer hyphen Verlag comma \ centerlineNew York comma{ [ 4 1 ] 995\quad periodBauer C . , Frink A . , Kreckel R . , Vollinga J , G i NaC is Not a CAS , \quad http : //www. gin ac . de/ . } open square bracket 6 closing square bracket .. Beardon A period F period comma Algebra and geometry comma Cambridge University Press[ 5 ]comma\quad CambridgeBeardon comma A . F 2005 . period, The geometry of discrete groups , Graduate Texts in Mathematics , Vol . 9 1 , Springer − Verlag , Newopen York square , bracket1 995 7 . closing square bracket .. Bekkara E period comma Frances C period comma Zeghib A period comma On lightlike geometry : .. isometric actions comma and rigidity aspects comma .. C period .. R period \ centerlineM .. a t h period{ [ 6 .. ] A\quad cad periodBeardon S ci period A . .. F P . a ,r i Algebra s 343 open and parenthesis geometry 2006 ,closing Cambridge parenthesis University comma 317 Press endash 32 , Cambridge1 period , 2005 . } open square bracket 8 closing square bracket .. Benz .. W period comma .. Classical geometries in modern contexts period .. Geometry of real\ hspace inner∗{\ productf i l l ..} [ spaces 7 ] comma\quad ..Bekkara 2 nd ed period E . comma , Frances C . , Zeghib A . , On lightlike geometry : \quad isometric actions , and rigidity aspects , \quad C. \quad R. Birkh a-dieresis user Verlag comma Basel comma 2 7 period \ centerlineopen square{M bracket\quad 9 closinga t h square . \quad bracketA .. cad Berger . S M c period i . \quad commaPa Geometry r i s period 343 II ( comma 2006 )Springer , 317 hyphen−− 32 Verlag 1 . comma} Berlin comma 1 987 period \ hspaceopen square∗{\ f i bracket l l } [ 8 1 0 ] closing\quad squareBenz bracket\quad ..W., Boccaletti\quad .. D periodClassical comma geometries .. Catoni .. F periodin modern comma contexts .. Cannata ... R\quad periodGeometry comma of real inner product \quad spaces , \quad 2 nd ed . , .. Catoni .. V period comma .. Nichelatti .. E period comma .. Zampetti .. P period comma .. 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Cnops J period comma Hurwitz pairs and applications of M dieresis-o bius transformations comma\ hspace Habilitation∗{\ f i l l } Dissertation[ 1 3 ] \quad commaCerej University eiras P . ,K $ \ddot{a} $ hler U . , Sommen F . , Parabolic Dirac operators and the Navier −− Stokes equations over of Gent comma 1 994 period \ centerlineopen square{ brackettime − 1 6varying closing square domains bracket , ..\quad CnopsM J period a th comma . \quad AnM introduction e t h o to d Dirac s \quad operatorsA ppl on manifolds . \quad commaSc i .. . Progress 28 ( 2005 ) , 1715 −− 1 724 . } in Mathematical Physics comma Vol period 24 comma \ hspaceBirkh∗{\ a-dieresisf i l l } user[ 14 Boston ] \quad commaChern Inc period S . comma− S . Boston , Finsler comma geometry MA comma is 2002 j period ust Riemannian geometry without the quadratic restriction , Notices Amer . open square bracket 1 7 closing square bracket .. Coxeter H period S period M period comma Greitzer S period L period comma Geometry revisited\ centerline comma{Math Random . SocHouse . comma 43 ( New 1 996 York ) comma , 959 1−− 9 67963 period . } open square bracket 1 8 closing square bracket .. Davis M period comma Applied nonstandard analysis comma John Wiley ampersand Sons comma[ 1 5 New ] \quad York commaCnops 1 J 977 . period , Hurwitz pairs and applications of M $ \ddot{o} $ bius transformations , Habilitation Dissertation , University ofopen Gent square , 1 bracket 994 . 1 9 closing square bracket .. Eelbode D period comma Sommen F period comma Taylor series on the hyperbolic unit ball comma Bull period Belg period Math period Soc period Simon Stevin 1 1 \ hspaceopen parenthesis∗{\ f i l l } 2[ 4 1 closing 6 ] \ parenthesisquad Cnops comma J . 71 , 9 An endash introduction 737 period to Dirac operators on manifolds , \quad Progress in Mathematical Physics , Vol . 24 , open square bracket 20 closing square bracket .. 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[ 23 ] Gromov N . A . , Contractions and analytic extensions of classical groups . Unified approach , Akad . Nauk SSSR Ural . Otdel . Komi Nauchn . Tsentr , Syktyvkar , 1 990 ( in Russian ) . [ 24 ] Gromov N . A . , Kuratov V . V . , Noncommutative space - time models , Cz e c h o s lov ak J . P hy s . 55 ( 2005 ) , 142 1 – 1 426 , he p - t h / 0 5 0 7 0 9 . [ 25 ] Herranz F . J . , Santander M . , Conformal compactification of spacetimes , J . Ph ys . A : M a th . G e n . 35 ( 2 2 ) , 6619 – 6629 , ma th - p h / 0 1 1 0 0 1 9 . [ 26 ] Herranz F . J . , Santander M . , Conformal symmetries of spacetimes , J . P h ys . A : M ath . G e n . 35 ( 2002 ) , 6601 – 6618 , m a t h - p h / 0 1 1 0 0 1 9 . [ 27 ] Howe R . , Tan E . - C . , Non - abelian harmonic analysis . Applications of SL(2, R), Springer - Verlag , New York , 1 992 . [ 28 ] Khrennikov A . Yu . , Hyperbolic quantum mechanics , Dokl . Akad . Nauk 402 ( 2005 ) , 1 70 – 1 72 ( in Russian ) . [ 29 ] Khrennikov A . , Segre G . , Hyperbolic quantization , in Quantum Probability and Infinite Dimensional Analy - sis , QP – PQ : Quantum Probab . White Noise Anal . , Vol . 20 , World Sci . Publ . , Hackensack , NJ , 2007 , 282 – 2 87 . [ 30 ] Kirillov A . A . , Elements of the theory of representations , Grundlehren der Mathematischen Wissenscha f − t en , Vol . 2 20 , Springer - Verlag , Berlin – New York , 1 976 . [ 31 ] Kirillov A . A . , A tale of two fractals , see h t tp : / / w ww . math . u p en n . edu / ∼ kir i l lov / M ATH 48 0 - F 0 7 / t f . p df . [ 32 ] Kisil A . V . , Isometric action of SL2(R) on homogeneous spaces , A d v . App l . C l iff o r d A lg e b r . 20 ( 2010 ) , 299 – 3 1 2 , a r Xiv : 8 1 0 . 3 6 8 . [ 33 ] Kisil V . V . , Construction of integral representations for spaces of analytic functions , Dokl . Akad . Nauk 350 ( 1 996 ) , 446 – 448 ( Russian ) . [ 34 ] Kisil V . V . , M o¨ bius transformations and monogenic functional calculus , Elec t r o n . Re s . A n n o u n c . Am e r . M a t h . S o c . 2 ( 1 996 ) , 26 – 33 . [ 35 ] Kisil V . V . , Towards to analysis in Rpq, in Proceedings of Symposium Analytical and Numerical Methods in Quaternionic and Clifford Analysis ( Seiffen , Germany , 1 996 ) , Editors W . Spr o¨ ßig and K . G u¨ rlebeck , TU Bergakademie Freiberg , Freiberg , 1 996 , 95 – 1 0 . [ 36 ] Kisil V . V . , How many essentially different function theories exist ?, in Clifford Algebras and Their Applica - tion in mathematical physics ( Aachen , 1 996 ) , Fund . Theories Phys . , Vol . 94 , Kluwer Acad . Publ . , Dordrecht , 1 998 , 1 75 – 1 84 . [ 37 ] Kisil V . V . , Analysis in R1,1 or the principal function theory , C o m p l e x V a r ia b l e s T h e or y A p p l . 40 ( 1 999 ) , 93–118,fu nct-an/9712003. [ 38 ] Kisil V . V . , Relative convolutions . I . Properties and applications , A d v . 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V period V period Kisil \noindent 44 \ h f i l l V . V . K i s i l Nohliner − t h - Holland Math . Stud . , Vol . 1 97 , Elsevier , Amsterdam , 2004 , 1 33 – 141 , m at h . F A / 2 0 82 4 9 . open square[ 44 bracket ] Kisil 23 V closing . V . , square An example bracket of Clifford .. Gromov algebras N period calculations A period with commaG iNaC Contractions, A d and v . analytic A pp extensions l . of classical groups \ [ \ r u l e {3em}{0.4 pt }\ ] periodC Unified l iff o rapproach d A l comma g e b r .Akad15 period( 2005 ) Nauk SSSR Ural period Otdel period Komi Nauchn period2 39 – Tsentr 269 , cs comma . M Syktyvkar S / 4 1 0 44 comma . 1 990 open parenthesis in Russian closing parenthesis period open square[ 45 ] bracket Kisil 24 V closing . V . , square Starting bracket with .. Gromov the N group period ASL period( R comma), Notices Kuratov V Amer period . V Mathperiod . comma Noncommutative \ hspace ∗{\ f i l l } [ 23 ] \quad Gromov N . A . , Contractions2 and analytic extensions of classical groups . Unified approach , Akad . Nauk SSSR space hyphenSoc . time54 models( 2007 comma ) , .. 1 Cz 458 e – c 1 h 465 o s , lov ak J period P .. hy .. s period 55 open parenthesis 2005 closing parenthesis comma 142 1 endash m a t h . G M / 6 0 7 38 7 . \ centerline { Ural . Otdel . Komi Nauchn . Tsentr , Syktyvkar , 1 990 ( in Russian ) . } 1 426 comma[ 46 ] .. he Kisil p hyphen V . V .t , Fillmoreh slash 0 – 5 Springer 0 7 0 9 – period Cnops construction implemented in G iNaC , A d v . A pp l . openC squareliff ord bracket A lg 25 e b closing r . 1 7 square bracket .. Herranz F period J period comma Santander M period comma Conformal compactification \ hspace ∗{\ f i l l } [ 24 ] \quad Gromov N . A . , Kuratov V . V . , Noncommutative space − time models , \quad Czechos lov akJ .P \quad hy \quad s . 55 ( 2005 ) , 142 1 −− of spacetimes( 2 7 ) , comma 59 – 70 J , c period s . M Ph S ys / period0 5 1 20 A 7 : 3 .. . 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