Title Oriented Circles in Non-Euclidean Space

Author(s) Nishiuchi, Teikichi; Kashiwagi, Hidetoshi

Memoirs of the College of Science, Kyoto Imperial University Citation (1920), 4(6): 273-303

Issue Date 1920-09-20

URL http://hdl.handle.net/2433/256564

Right

Type Departmental Bulletin Paper

Textversion publisher

Kyoto University Oriented Circles 1n Non-Euclidean Space,

BY

Teikichi Nishiuchi and Hidetoshi Kashiwagi.

(Received, April 26, 1920)

CHAPTER I

§ 1

Coordinates of an Oriented Circle in Non-Euclidean Space.

Circles in euclidean space have been studied already by Laguerre, Koenigs, Stephanos, Cossert, Castelnuovo and Coolidge. But the study of circles in non-euclidean space has not been taken up by anybody. We will discuss the oriented circles by a method analogous to that of R. Bricard0 l in his paper 'Geometrie des feuillets' which is the 2 generalization of line-geometry · and that of Study. C J And we shall consider a circle in non-euclidean space as the intersection of a sphere and a plane. Thus any sphere and any plane can define a circle, and without any loss of generality, we may consider a circle as the intersection of a sphere and a plane passing through the center of the sphere. According to the definition, a circle in non-euclidean space may be represented by the equations of the form :

1 See B. Bricard, Comptes Rendus, Tome. 150, 1910, p. 1586. See also Nouvelles Annales de Mathematiques, series. 4, Tome. x. 2 See Study, Geometrie der Dynamen. 274 Teikichi Nishittchi and Hidetoslti Kashiwagi

(a,z)2=cos2-~-(z,z), k (b,z)=o with the condition (a,a)= 1, (a,b) =o, (b,b)= 1,

where (a) is the coordinates of the center of the sphere, (b) that of the plane of the circle, r the radius of the sphere and k the measure of curvature of the space. The center of the circle coincides with the center of the sphere. We shall call the absolute pole of the plane of the circle absolute pole of the circle and the sphere main sphere of t!te circle. The absolute polar plane of the center of the circle shall be called central plane of the circle and the intersection of the central plane of the circle and the plane of the circle plane-a.xis of the circle. The line perpendicular to the plane of the circle passing through the center of the circle shall be called space-axis of tlze circle. The axial coordinates of the plane-axis are

and the radical coordinates of the space-axis are

and these two lines are evidently absolute polars to each other. Next, we shall define a polar sphere as the sphere which is the envelope of the absolute polar plan~s of the points on a given sphere. The equation to the polar sphere of a given sphere

(a,z)2 =cos2____1'__(x,x). k IS (a,x ) .,·=sur-- ."r(x,x. ) k

And we shall define the intersection of the plane of a circle and the polar sphere of the main sphere as the polar circle. Now, we shall define the foci o/ tlte circle as the centers of the null-sphere passing through the given circle in analogy to Laguerre's Oriented Circles in Non-Euclidean Space 275 definition1 in euclidean space. According to this definition, we have two foci whose coordinates are

(a±isin~b), where i= ✓ -1, and the coordinates of two foci of the polar circle are

(a±i cos~ b), where i= ✓ -=---i":"

The foci may be considered as the intersecting points of the isotropic lines passing through the circle. We shall define an oriented circle as a figure formed by a circle and the two foci of the polar circle in that order, calling one the first and the other the second. When the roles of the two foci have been interchanged, the resulting oriented circle shall be called to be the opposite of the original one. A circle whose radius .!!_k is its own opposite. 2 \Ve shall define the common points of two oriented circles which lie on one of the radical planes of the main spheres

(a,x)cos-r' - (a',x) cos-=or k k their points of intersection, where the equations of the two main spheres are (a,x)2 =cos2~(x,x), (a',x) 2=cos2~(x,x). k · k

Hence, two oriented circles can not intersect at more than two points. When two oriented circles are coplanar, they will intersect at two points and the line joining these points of intersection is given by the equation (a,x)cos-r' -(a',x)cos-=o,r k k where the equations of the two coplanar circles are

1 See Laguerre, •Sur l'emploi des imaginaires dans la Geometrie de l'espace,' Nouvelles, Annales de Mathematiqnes, series. 2, vol. XI, 1872. 276 Teikiclzi Nlshiuchi and H1detoshi Kashiwagi and cosine of the angles of intersection will be

r r' t1> (a, a')-cos-cos­ k k cos8= • 1· • r' Stn-Stn- k k

Let (a 0 , ai, a2, a3) and (b0 , bi, b2, b3) be Weirstrass' coordinates of the center of a circle and the pole of its plane, and we put

a_a=T, iia .

then (a,a)= r, (a,b)=o, (b,b)= 1. We shall call

the coordinates of an oriented circle. Now, we adopt the following notation: . rb. rb . rb. rb a , ai, a , a , zcos- , icos- i, zcos- , zcos- a 0 2 3 k 0 k k 2 k

where we always take a0 >o, b0 >0, if a0 =0 (or b0 =0) then a 1 >0 (or b1 >o) and so on, and

v(a,a)= I, v(b,b)=I. It is evident that all the determinants of the matrix

Po P1 Pt. jJ3 II II q. qi q2 qa are not equal to zero, for if all of them vanish, then

(i=O, I, 2, 3) which is absurd.

1 See Coolidge, Non-Euclidean Ge01netry, p. 137. Oriented Circles in Non-Euclidean Space 277 We see that there exists a relation (p, q)==-Poqo+ P1q1 +P2q2 + Psq3=0, which is analogous to Plucker's identity of the line-coordinates. So we shall call the coordinates ( { ) Plucker's coordinates ef the oriented circle. From the definition of the polar circle, it is evident that an oriented circle and its polar circle are concentric and coplanar, but their radii differ by _!!_k, Hence, J%cker's coordinates of the polar 2 circle of an oriented circle whose coordinatas ( { ) are

.. rb .. rb .. rb .. rb a0 , ai, a2, a3, zszn- "' zszn 1, zszn- 2, tszn- 3 k k k k and we shall denote it by the notation

Po, Pi, P2, Ps, 'iio, qi, "§z, "§s= ( : ).

We can easily .see that there exists a following relation between Plucker's coordinates of an oriented circle and its .polar circles:

(p,p)+ (q, q) +( p,p) + (q, q)= I. A circle whose radius is zero is called a null-circle. The co­ ordinates of a null-circle satisfy the following condition : (p,p)+(q, q)=o. Again we put Xo=Po+qo, Xi=Pi+qi, X2 P2+q2, Xa /a+qs, oX =i(Po-qo), 1X =i(Ji-qi), 2X =i(}2-q2), aX =i(Ps-qJ), where i= ✓ =r-" Then we notice that (Xi)=X.: X1: X2: Xa and (;X)=.X: 1X: 2X: sX 278 Teikichi Nishiuchi and Hidetoshi Kashiwagi are respectively the coordinates of the foci of the polar circle of the given oriented circle. . The oriented circle may be represented by '

~o, ~1, X2, Xa, 0 X, 1X, 2X, aX =C:j) and Plucker's identity becomes as follows: (X, X)=(Xj, XJ)+(,X,j'~)=o, which is analogous to the Klein identity of the line-coordinates. Hence, we shall call ( : ) Klein's coordinates of the oriented circle. So the coordinates of the polar circle of the given oriented circle are represented by where xj 7'i+qj, JX =i(pi-qJ), (i= ✓ -1, j=O, I, 2, 3). There exist the following relations between (XJ), (;X) and (Xj), (1X): ( XJ, XJ)+( XJ, Xi)= 1, cx,jx )+GX, jx )=-I. If the given oriented circle is null, then we have ( xj, Xj)=GX,jX )=o.

§ 2 Relations between two Oriented Circles. Suppose two circles<1> with Plucker's coordinates ( ~ ) and ( {;) are given. If p,=Pl (i=o, 1, 2, 3), then they are concentric. If ~=~ (i=O, I, 2, 3), then they are coplanar, or copolar. If (p,p')=o, then their centers are orthogonal, or the central planes cut ortho­ gonally to each other. If (p, q')=o,

1 For the sake of brevity, we shall call, here in after, an oriented circle a circle. Oriented Circles in Non-Euclidean Space 279 then the center of one circle lies on the plane of the other, or the center of one circle and the pole of the other circle are mutually orthogonal. If (q, q') =O, then the planes of the circles cut orthogonally, or the poles of the circles are mutually orthogonal, when the radius of both of the circles is no t equa 1 to _!!___k, 2 If (q, q)=(q', q'), then the two circles are equiradii. If ( P, q') + (p', q) =O, then we have . r . r' Sl/l- szn­ k k ' --1, cos ( a/), or . r r' sin­ sin- k k -I, d d' sin- sin- k k where ab') . d ( a'b) . d' cos ( k =stnk, cos k =sink.

This relation is elegant, this means that one set of the four points on the line of centers cut by the two main spheres of the circles and two other spheres of which each is described with a center, the center of one circle, and touching the plane of the other forms a harmonic group. If (p ,q')+(p', q)=o, then the angles formed by the radical plane of the main spheres

(a, .:t-)cos-r' - (a', x)cos-r = o k k and the planes of the circles are equal. If two non-coplanar circles whose radii are not equal to ~k are ;J cospherical, then, first, their space-axes should cut each other. Let the coordinates of the point of the intersection of their space-axes be (a+lb) or (a'+l'b'), then Teikichi Nishiuchi and Hidetoshi Kashiwagi

a0 + A b0 a1 + A bi _ a2 + Ab 2 _ a3 + Ab3

ao' + ..t'b0' a/+ ..t'bi' - az' + ).'b/ - as'+ ..t'bs' and r r COS- cos- k k

Hence, eliminating A and A1 from the above equations, we get the following conditions that the two non-coplanar circles are cospherical

ao a1 a2 fl3 bo bi b2 ba ao' a/ a/ aa' =o,

b'0 bi' bl ba' . and all the determinants of the matrix

r' r r' r r' r a cos- - a 1cos- a cos-- a11 cos- ...... a cos-- a 1cos- 0 k 0 k 1 k k 8 k 3 k

are equal to zero. Or in Plucker's coordinates

I P q P' q' l=o and all the determinants of the matrix

I Pov(q',q')-Po'✓ (q,q)P1i/(q',q')-Pi'1I (q,q) .... ·•Pa-v' (q',q')-Ps'i/(q,q) , qo ql • .,,. q3

are equal to zero. If two circles intersect at one point only, the point of the inter­ section should lie on the radical plane of their main sphere

r' . (a, x)cos - -(a', x )cos___!:_= o k k Oriented Circles in Non-Euclidean Space 281 and the planes of the circles

(b, z)=o, (b', z)=o.

Hence, when they are not coplaner and cospherical, the coordinates of the point of intersection are given by the ratio of the determinants of the matrix

If two circles intersect at two points, then the radical plane of their main spheres

(a, z)cos--(a',r' z)cos-=or k k

and the planes of the circles are coaxial, hence all the determinants of the above matrix are equal to zero.

§ 3

On Circles and Their Main Spheres.

Theorem. The necessary and sufficient condition that the two main spheres of two coplanar circles whose coordinates ( ! ) and

( {'. ) should touch each other is that

(p,p')+(q, q') ✓ ( p,p) + (q,q)-.I (p',P') + (q',q')

Theorem. The necessary and sufficient condition that the two main spheres of two coplanar circles should cut orthogonally is that

(p, p')+(q, q')=o. 282 Teikichi Nishiuchi and Hzdetoshi Kashiwagi

Theorem. If the planes of the two circles are mutually ortho­ gonal, the necessary and sufficient condition that their main spheres. should cut orthogonally is that

Theorem. If the centers of two circles lie on the intersection of the planes of the circles and their main spheres cut orthogonally, then the first main sphere passes through the foci of the second circle and the second main sphere passes through the foci of the first. Theorem. If the two circles are concentric and

(q, q')2= I, then the foci of the two circles are mutually orthogonal. Theorem. If the two circles are concentric and their planes are mutually orthogonal, then the distances between the foci of the two circles are equal to zero. If (p,j/) + (q, q') I' ✓ (P,P) + (q,q) ✓ (P',p') + (q',q') then

( a a ') -cos-cos-r r' ' k k . r . r' (b, b') szn-sm- k k or cos 0= cos

(p,p') + (q, q') -;:====:c--~=====-=I ✓ (p,p) + (q,q)1 I (p',p') + (q',q') ' then the angle between the two planes of the circles and one of the angles formed by their main spheres are equal. on·ented Circles in Non-Euc#dean Space

§ 4 Common Orthogonal Sphere of two Circles. When a sphere is orthogonal to every sphere through a circle, we shall call the sphere orthogonal to the circle. Suppose a sphere R (ax)2=cos2-(a, a)(x, x) k be orthogonal to a circle whose Plucker's coordinates are

. r b . r b . r b . r z.c a11 ~, as, zcos- 0 , 1cos - 1, zcos - s, zcos -u3, k k k k then and (a, a) R r cos-cos-. ✓ (u, a) k k Hence, we get the following theorem. Theorem. The necessary and sufficient conditions that a sphere be orthogonal to a circle are that the center of the sphere should lie on the plane of the circle and the given sphere should cut the main sphere of the circle orthogonally. Theorem. If a sphere is orthogonal to a circle, then the sphere will pass through the foci of the circle. Now we will find the common orthogonal sphere of two non- cospherical and non-coplanar circle with Plucker's coordinates ( t ) and \, P')t/ . Let the equation of the required sphere be (a, x)2=cos2_!i_(a, a)(x, x), k then (a, b)=(a, b')=o, (a, a) R r cos-cos- 1/(a, a) k k and (a', a) R r' COS-COS-. v(a, a) k k Teikichi Nishiuchi and Hidetoshi Kashiwagi

Therefore, the coordinates of the center of the common orthogonal sphere are

t a cos_!'_- a'cos___!__ b b' l(i) k k or

where p, p' are proportional factors and i= o, I, 2, 3. And the radius of the common orthogonal sphere is given by the formula

R (a, a) (a', a) cos k

By this discussion, we see that the common orthogonal sphere will pass through the foci of the two circles and the center of the sphere lies on the intersection of the planes of the circles and the length of the tangents from the center of the sphere to the two given circles are equal to the radius of the sphere. If two non-coplanar circles intersect at one point, then their point of intersection will coincide with the center of the common orthogonal sphere and the sphere will be a null-sphere. So the condition that two non-coplanar circles should intersect is that

(a, a) (a', a) - I,

or

1 This notation means the coordinates (") are the ratio of the determinants of the matrix Oriented C£rcles z'n Non-Euclidean Space

a. al a2 aa 2 bo b1 b2 ba

a'0 a'i a'2 a's

b' 0 b'i b'2 b's

r' , r r' , r r' , r' 2 aoCOS--a .cos- a1COS--a 1COS- a3COS--a 3COS- k k k k k k =

i. e

I o (a, a') (a, b') o I (b, a') (b, b') (a', a) (a', b) I o (b', a) (b', b) o I

2r' r r' r r r' cos --- z(a, a 1 )cos-cos-+cos2 - -(a', b)cos-k (a, b')cos-k - k k k k

-(a', b)cos.!:_ (b, b') k

(a, b')cos.!!_ (b', b) k or

I O (p,p') (p, q') 0 I (q,p') (q, q')

(jJ 1,jJ)(p', q) I 0

(q',p) (q', q) 0 I

(q',q')- 2(p,p') ✓ ( q,q) ✓ (q',q')+(q,q) -(P',q) ✓ (q,q) (p,q') ✓ (q',q')

= -(jJ', q) ✓ (q, q) I (q, q') (q', q)

From this condition and that in §2, we see that when two non­ coplanar circles intersect at two points they are cospherical. A system of circles which is orthogonal to a sphere shall be called · a normal system of circles. The planes of the circles of this system pass through a common point which is the center of the 286 Teikichi Nisliiuclti and Hidetoshi Kashiwagi sphere, and the length of every tangent from this point to the circles of this system are all equal to the radius of the sphere. Moreover the points of contact of these tangents and the foci of the circles lie on the sphere.

§ 5 The Angle Between Two Circles at Their Points of Intersection. We can define the angle between two circles at their points of intersection by the angle of the tangents to the circles at that point. Let (y) be the coordinates of a point of intersection of two circles with the coordinates ( P ) and ( The plane-coordinates of q P: q ). the tangent planes to the main spheres at that point are

(i=O, I, 2, 3) where A and ).' are proportional factors. Hence, the axial-coordinates of the tangents to the two circles at the point (y) are

ui Uj I rr.1=l b. b1 '

u'i .1 n'.1=/ u'•1 b' i b'1 .

The cosine of the angle formed by them (i. e. the cosine of the angle of the circles at the point of intersection) will be

~'TCij r.' iJ cos (} = ----cc-~----''---- v' ~r. tf i/Ir.'1i (u, u') (u, b') ) I (b, u') . (b, b') : ✓ I (u, u) (u, b) I· jj (u', u') (u', b') I (b, u) (b, b) 1 (b', u') (b', b') Oriented Circles £n Non-Euclidean Space

But r r' r' , r (a,y)2=cos 2-k(y,y), (a',y)2=cos2-(y,y), (a,y) cos-=(a ,y) cos-, k k k

(b,y)=o, (b',y)=o, (a, a)=(b, b)= I. Hence, we have r r' (a, a') (b, b')-(a, b') (a', b)-cosk cosk (b, b')

cos{}= .J . r . r szn- sin- k k

(P,P') (q, q')-(p, q') (p', q)-(q, q') ✓ ~0q) ✓ (q', q') ✓ (q, q) ✓ (q', q') ✓ (P,P) +(q, q) ✓ (P',p')+(q', q') The condition of contact will be cos0= + r, that is

(a, a') (b, b')-(a, b') (a', b)-cos_!!..__cos_!:!__(b, b') =sin_!:_s£n_!!..__ k k k k and of orthogonal intersection

~ a ' aj I• I a' i a'.i I= cos-cos-cos¢r r' \ b. bj b'. b'j k k or

(i,j= O, I, 2, 3), where ¢ is the angle between the planes of the circles. If the given two circles are coplanar, then

r r' <1l (a a')--cos-cos- ' k k cosO=------. r . r szn-szn­ k k

1 This is the formula which we introduced in ~I. See also Coolidge, Non-Euclidean Geometry, p. 137. 288 Teikichi Nishiuchi and Hidetoshi Kashiwagi

Two circles cut orthogonally to each other, when one of the following conditions is satisfied : ( i ) Two main spheres cut mutually orthogonally and one of the centers of the circles lies on the plane of the other. ( ii ) Planes of the circles are mutually orthogonal and one of the centers of the circles lies on the plane of the other. ( iii ) One of the main spheres is orthogonal to the other circle. ( iv ) The space-axes are mutually orthogonal and the planes of the circles are orthogonal to each other.

If ( { ) and ( {; ) are the coordinates of two circles I' and I',' the expression

cos( a;' ) cos( bt' )- cos( a;' ) cos( a; )- cos( b:' ) cos ~ cos ~ shall be called the mutual power of the circles in analogy to the de­ finition of that in the Spherical Trigonometry<1> and denote it by P( I', I''). When the circles intersect, their mutual power is equal to the product of the sines of their radii divided by the measure of curvature and the cosine of their angle of intersection. When the circles do not intersect, their mutual power is not susceptible of this geometrical interpretation. In special case other interpretations can be found; thus if both circles be in the same plane, their mutual power is cos( a;') r r' -cos-cos-, and moreover in this case, if both circles be null their k k mutual power is - 2 sin 2½, ( ~a' ).

s 6

2 Involution and Bi-Involutionc > Two circles shall be said to be in involution when each is orthogonal to a sphere through the other.

l See Todhunter and Leathern, Spherical Trigonometry. 2 This name is adopted in analogy to that of euclidean splCe given by Koenigs, See his remarkable article 'Contributions il la theorie du cercle dans l'espace', Annales de la Faculte des Sciences de Toulouse, vol. ii, 1888. Oriented Circles in Non-Euclidean Space

Theorem. The necessary and sufficient condition that two circles should be in involution is that their mutual power is equal to zero. Theorem. If the space-axes and the planes of two circles are mutually orthogonal, then they are in involution. Theorem. When we construct two spheres through two non­ coplaner circles, each center lies on the plane of the other, if the first sphere pass through the foci of the second circle, then the second sphere will also pass through the foci of the first, and they are in involution. Tlteorem. If two circles intersect and are in involution, then they cut orthogonally to each other. Next, we shall define two circles as being in bi-involution when every sphere through one is orthogonal to the other. Theorem. The necessary and sufficient condition: that two non­ coplanar and non-cospherical circles should be in bi-involution are that

(p, q')=o, (p', q)=o, (q, q')=o, and

(p, p')+ v(q, q) ✓ (q', q')=o, that is ( i ) The centers of the circles lie on the intersection of the planes of the circles. ( ii) The planes of the circles- are orthogonal, ( iii ) Their main spheres cut orthogonally to each other. Theorem. When two circles intersect and are in bi-involution, then they cut mutually orthogonally.

§ 7 Pseudo-Distances and Angles of two Non-Intersecting Circles. Let ( t ), (t; ) be the coordinates of two non-coplanar and non­ cospherical circles I' and I", then the coordinates of the center of the common orthogonal sphere are

t acos__!!__-a, cos~ b b' I k k Teikicki Niskiucki and Hidetoslzi Kaskiwagi where A is the propotional factor. Consider the polar lines of (a), the center of the common orthogonal sphere, with respect to the given circles and denote them by l, l'. We shall mean the pseudo-distances of the two non-intersecting circles the distances of the lines l and l'. As the polar plane of ( a) with respect to one of the main spheres is given by the equation

(a, a) (a, x)-cos2_!_ (a, x)= o, k the axial coordinates of the polar lines l and l' are

and r' (a,), , a a i-cos 2r' -a, (a', a) a'i-cos2 ka, k b'-• b'3 Now, if we put

(i, j=O, I, 2, 3), then we get

As (a) are the coordinates of the center of the common orthogonal sphere, we have the following relations:

r R r' R = cos-cos-, cos-cos- k k k k where R is the radius of the sphere. Hence, we have Oriented Circles in Non-Euclidean Space

where P(I', I'') is the mutual power of the two circles and ¢ is the angle of the planes of the circles. And

(l Il') = l'l,j l'kl (i, ;: k, l=o, I, 2, 3)

=(a, a) (a', a) (X,IX')-(a', a) cos2___!:_(~'l,)-(a, a)cos2_i_(X, [, ') k k

-- r r' R -- r r' R =(a, a) v'(rL, a) cos-cos-cos3--(a, a)./ (a, a) cos-cos_cos- k k k k k k

= - (a, a) v'-() a, a cos-cos-cos-szn r r' R.R2- k k k k

a. a1 a2 aa = -(arz) b. bi b2 bs cos-cos-sinr r' -2R - a'. a'1 a'2 a's k k k b'. b'1 b'2 b'a

Let d1 and d2 be the distances of the lines l and l', then the square of their moment< 1l will be

(l \ l')2 c2) szn-szn-=-2r!1 ·2d2 k k ( Xtt) . ( Xt':j)

1 See D'Ovidio, 'Studio sulla geometria proiettiva,' Annali di rnathernatica, vi, 1873. 2 See Coolidge, Non-Euclidean Geometry, p. 112. Teikichi M'shiuchi and Hidetoshi Kashz'wagi ., R I a b a' b' I tan4k =( sin·-+•or cos--tan--or .R)( sin·-+•or' cos 2r'-tan·- .R) k k k k k k and sin2-d1- + sin2 d2 k k

4 2 laba'bf tan f-{P(I:I'')+cos¢cos; cos: tan 1r (t) ~r---~-r--o-R=-c-)----c(-•--r~'---2-r'~-2-R=---c)- =I+~~(---szn 2- + co:,-tmr- szn 2- + cos -tan - k k k k k k

From these equations, we can find the pseudo-distances of the circles. And the square of the commoment< 2> of the lines l and l' is

2 p ( I', I'')+ cos¢ cos__!___ cos__!"_tan2 R } { k k k

We shall call the momment (commoment) of the lines land l' the pseudo-moment (pseudo-commoment) of the circles. When the circles intersect or their space-axes cut each other, their pseudo-moment vanishes, and when their mutual power is equal to zero and the planes of the circles cut orthogonally then their pseudo-comment vanishes. We shall call the angles of the lines l and /1 the pseudo-angles of the circles.

Let 01 and 02 be the angles of the lines l and l', i.e. the pseudo­ angles of the circles, then

2 1 4 R a b a b' tan - . "(} . "(} I 1 k szn· 1 sur 2 R ) r' r' R , = ( 2 2 2 2 2 sin __!_+ cos __!__tan - (sin - + cos2-tan -) k k k k k k

I, 3 See Coolidge, Non-Euclidean Geometry, p. II2. 2 See D'Ovidio, 'Studio sulla geometria proiettiva,' Annali di mathematica, vi, 1873. Oriented Circles in Non-Euclidean Space 293 and

When the circles intersect, then {P(I',I'') }2 cos20 ·21' .• r' szn-szn-- k k This is the formula which we reduced in § 5. Two circles shall be said to be pseudo-paratacticC1l when their pseudo-distances and pseudo-angles are all congruent.

CHAPTER Il.

§ 8

Other Coordinates of Circle. Next, we put

where (a 0 , ai, a2 , a3), (b 0 , bi, b2, b8) and r have the same meaning as in§ I. ,

We can take (X 0 , X1, X2, Xa, Xo1, X,2, Xoa, Xsi, X12)=(X) as the coordinates of a circle. This is analogous to the coordinates of 'the Soma of the second sort' given by Study.c2l So we shall call

1 This word is adopted in analogy to the word ' paratactic.' See Study, ' Zur Nicht­ Euclidischen und Linien-Geometrie,' Jahers-bericht der deutschen Mathematikervereinigung, xi, 1902. 2 See Study, Geomertie der Dynamen. 294 Teikichi Nishiuchi and Hidetoshi .Kashiwagi this coordinates Study' s coordinates of the circle. It is evident that there exists the following relation :

When we have two cfrcles I' and .I" with Study's coordinates(%) and (,X;'), then

p (I', I'1)=(2,X;,;,%',;,) + 2,X;./lf/ij, (i/=O, I, 2, 3). If we adopt the following notation

then P(I',I'')=[X, X']. If

then they are coplanar. If X ~+Xi+ X ~+ X ~=X'~+X'i+ X'~+ ~'~, then they are equiradii. If /%,;,j=X'.j, (i,j=o, 1, 2, 3), then they have the same space-axis. If (,X;., X'.)=XoX'o+ X1X'1 + X2,X;'2+ Xs~'s=O, then the planes of the circles cut mutually orthogonally. If (x;.j, X'.j)=2X.jX'.j=o, (i,j=o, 1, 2, 3) then their space-axes are mutually absolute polars. If

- th~ their ~spac~~a~es cut each ~ther. If "'(X:tj I X'.j)=(,X;.j, X\1~=0, , then their space-axes cut orthogonally to. each other. If Oriented Circles in Non-Euclidean Space 295

2 (X.,i I X.,i)=[(Xij, J(;ij) (J(;',u, X'.j)-(J(;ij, J(;\J ] =o, then their space-axes are either parallel or pseudo-parallel. If they intersect each other, their angles of intersection is given by the formula :

cos0

If they cut orthogonally or their mutual power is equal to zero, then or hence (J(;.,i, X\i)2-(Xi, X,'i)2 o. (Xij, ~ij) (.%\3, ~,ij) Therefore, the commoment of the space-axes of the circles is equal to r r' cos -cos -cos<}, k k . where

Let di and d2 be the distances between their space-axes and 01 and 02 their angles, then

and

As a special case, if the space-axes be paratactic and equiradii, then and

Similarly, if their polar circles cut mutually orthogonally, then

2 di 2 d, • 2 r • 2 r' ·· ·· '• , , cos -cos-" =szn -szn --cos' 't'·1 k k . k k If the circles are concentric, then Teikichi Nishzitchi and Hidetoshi Kashzwagi (Xii, X'ii)=cos cp. The condition that they should be in involution is that

and that of bi-involution is that

and

In this case, the square of the moment of their space-axes is equal to d1 • d. • r r' • aa') sm• 2Tsm 2 k- = r -cos"Tcos 2k= szn 2( T , i.e. when they are in bi-involution, the square of the moment of their space-axes is equal to the square of the sine of he distance of their centers divided by the measure of curvature. If we take two systems of seven circles with Study's coordinates

and form the product of the two vanishing determinants,

~ai) ~g) ~~~) xg) ~w xw o X~il Xa:i X bi) XgJ X ~il xg> o

and

9/ (1) 91 (1) 9/ (1) 9/ (1) 9/ (1) 9/ (1) 0 Jft Jm Jm J~ JU JU 9/ (2) 9/ (2) 9/ (2) 9/ (2) 9/ (2) 9/ (2) 0 J01 J02 J 03 J23 J31 J12

01 (7) 9/ (7) ,W (7) 9/ (7) 9/ (7) Jl/ (7) 0 ~m JOO Jm J~ Jfil JU then we have the following identity : Oriented Circles in Non-Euclidean Space

d d' d d' . d d' cos--1!!.. cos--2!!. cos-..E!.. cos_Ef_ cos__}J!_ cos_E!_ =O, k .k k k k k d. d' d. d' d. d' cos_E!_ cos_Jt!_ COS _El_ COS_'!.?!_ cos 271 cos_!!!_ k k k k k k

cos___B!_d cos---5}!_d' cos d121 cosd'121 cos__II!_d cos____II!_d' k k k k k k where d.j, and d'i,jl are the distances of the space-axes _of the circles with coordinates ( x; C•)) and ( :f (jl). If two sets of circles coincide respectively, then

d d' d d' I cos_E_ cos-E. cos__I!_ cos___E_ = o, . k k k k

d d' 21 d. d' cos~ cos- I cos__JJ__ cos-..E. k k k k ......

d d' d12 d'., cos---1!__ cos---.!!. cos- cos--•· I k. k k k where ~j and d'ij are the distances of the space-axes of the circles with coordinates (~Cil) and (~

I

. r2 r1 ,,, coS~OS-COSy21 ...... COS-COS-COSy27r2 r1 ,,, k k k k ...... ·- ...... " ......

...... ~ ...... ~ ...... r r1 ,,, r r2 ,,, COS-COS-COSy717 COS-COS-COSy7i7 ...... I k k k k that is r. cos-r 1 cos-t k k or Teikichi Nishiuc/zi and Hidetoshi Kashiwagi

I --- cos¢ ...... cos

cos

• d21 • 27 SZ1Z-SZ1Z-d'21· 0 sin!!E_sin d' k k k k

szn-szn-· d,1 · d'n szn-szn-"· d,2 · d'1° .. .. •• .. . o k k k k If they are in bi-involution to one another, then

0 sin(a:2) sin(af1) =O,

1 sin(aka ) 0 ...... sin(af1)

0 where sin( arj) is the sine of the distance between the centers of the circles with coordinates ( X Cil); ( X (jJ) divided by the measure of curvature. If (X) be the coordinates of the polar circle of a given circle with coordinates (,ll:f;), then Oriented Circles in Non-Euclidean Space 299

§ 9

Cospherical Circles.

If a circle with Study's coordinates (X) lies on a sphere

then the coordinates of the center of the circle is a;,-(b, a) b;, -,,II-(b, a) 2 and r I R cos-= · cos-, (i=o, r, 2, 3). k 1/I-(b, a)2 k Now, we put Xo, X1, X2, ,X3, Xo1, Xo2, Xos, X2s, Xs1, X12 ·. Rb. Rb . Rb. Rb =z cos- , i cos- 1, z cos- ; z cos- s, k 0 k k 2 k

ao a1 I Iao a2 l l ao as l l a.2 as Ias a1 l j a1 a21 Ibo b1 , bo b2 , bo bs , b2 bs bs b1 , b1 b2 . If two circles I and I'' lie on the same sphere

(a, :i:)2 = cos2 R (.r, x), (a, a)= I, k then the angles of the intersection is given by the following formula :

(a, a1)(b, b')-(a, b')(a, b')-,(b, b')cos-cos-r r' k k cos0= . r . r' sin-sin- k k

(b,b')-(a,b)(a,b')- (b,b') cos2 R k

And their mutual power is 300 Teikichi Nishiuchi .and Hzdetoshi Kashiwagi

(b, b')-(a, b) (a, b')-(b, b') cos; R k p (I',I') =------;:==;:==::---:==:=;:==:,:;--­ VI - (.b, u.J2vr-(b', a)2

If the planes of the circles pass through the center of the sphere, then they are equiradii and where ,P is the angle between the planes of the circles. If [X, X]=[X', X'] then they are equiradii. If

or

then one of the centers 0 of the circles is on the plane of the othere. If

then their centers coincide with that of the sphere (they are concentric), or they are coplanar.

§ 10 On the Mutual Power.

Let there be two systems of circles, each consisting of eleven circles, namely I;_, I's, I's, ...... , I;.1, and II1, Il2, Il3, ...... , Il11, whose 1 2 11 1 2 Study's coordinates are (XC J), (~C l), ... , (XC )) and (:yC )), (!:JC >), ... , ( f!f Clll), respectively. Form the product of two vanishing determinants Oriented Circles in Non,-Euclidean Space 30_1

1 Jl;;~l) x~1) xo) X (I) Jl;;Cl) xoJ Jl;;(l) xc'i0 X1 i ti! ,2 03 xg) 31 12 0 x~2) 2 x~2) 2 Jl;;C2) Jl;;C2J Jl;;C2) X c2i xc2) xc2) X1 l xi ) 01 02 03 23 31 12 0

and

1 flj~l) 1 flj (1) yc1J yr1J flj'l) flJ bi) Y1 ) Yi ) 01 flJW o3 23 31 ~Jg) 0 2 flj ~2) yc2J yc2) yc2J ff (2) yc2J flJ a2) Yi i flj~2) 01 02 03 23 31 flj~) 0 ...... ····················••t.••····· .. ···················

flf (11) fl/ CU) 9/ lll) fl/ (11) flf (11) 9f Cll) 9/ (11) 9f (11) 9f (11) flf CllJ O -;JO -;11 ;;/2 -;:13 -;:101 -;:102 -;]03 ;;123 ;:131 -;:112 ·and we get immediately following result analogous with Frobenius and Darboux's identity. Cl)

P ( I'i, Il1) P (I'i, fl2) P(I'i, fl11) =o.

P(½, Il1) P(½, Il2) P (I;, ll11) (1)

If each circle of one system intersect all the circles of the other system, the angle of intersection of the circles I',,, anc II,, being re­ presented by (m,n'), we derive the relation : .J

1 1 1 COS (I, I ) COS ( r°, 2 ) COS (I, I I ) = 0. cos ( 2, r') cos (z, z') cos (z, u') (2)

cos (II, 1') cos (II, 21) ••••••••• cos (rr, u') If the first ten circles of the first system are cut orthogonally by a same circle K', and if we take K' to be the eleventh circle of the second system, then the first ten elements of the last column vanish in this formula. Hence we get the following relation between the angles in which a system of ten circles, possessed of a common orthogonal circle, are cut by any other ten circles :

1 See G. Frobenius, 'Anweudungeu der D·eterminanten Theoric auf die Geometrie der Masses,' Crelle's Journal, vol. LXXIX, 1875, p. 187. See also Cayley, Cambridge Mathe­ matical Journal, vol. II, 1841, p. 267, or Collected Works, vol. I, p. I. See Darboux, 'Groupes de points de cercles et de sphares,' Annales de l'Ecole Normale, Series, 2, vol. i, 1872. 302 Teikichi Nishiuchi and Hidetoshi Kashiwagi

1 cos ( r, 1 ) COS (I, 2') ...... cos (r, ro') =o. cos (z, 1') cos (2, 2 1) ...... cos ( r, 10')

1 cos (ro, r') cos (ro, 2 ) ...... cos (ro,ro')

If K be a circle which cuts 111, 112 , ...... , 119 orthogonally, and if K' cut K orthogonally, K may be taken as the tenth circle of the first system. Then we get the following relation between the angles of inter­ section of the two systems of nine circles, whose respective orthogonal circles cut at right angles :

cos (r, 1') cos (1, 2') ...... cos (r, 9') =O. 1 cos (2, r') cos (2, 2 ) ...... cos (2, 9')

1 cos (9, 1') cos (9, 2 ) ...... cos (9, 9')

If the second system of nine , circles coincide with the first re­ spectively, then

r cos (r, 2) ...... cos (r, 9) =o. cos (2, r) r ...... c;s (2, 9) (5)

cos (9, 1) cos (9, 2) ......

In the relation (1), let the two sets of circles coincide, then it appears that any eleven circles satisfy the relation :

sin 22 k pen, Fi) sin2 r2 k (6) ·························································

sin2_!_'-g_ k

Thus the condition that the first ten circles are cut orthogonally by the eleventh is that Oriented Circles in Non-Euclidean Space

sin2i =0, k P(~, I'i_) sin2 r2 k

•••••••••••••••••••••••••••• •·&-• •••••••••••••••• " •••••••••

And the condition that the first ten circles touch the eleventh is that (when they intersect one another)

I cos (r, 2) ...... cos (1, 10) I =O,

...... COS (2, 10) I ...... cos (10, 1) cos (10, 2) ...... I I I I ..... , ...... I I or

2 (1, 2) sin2(Ii IO) 0 sin ~ ...... =O. 2 2 2 (2, I) sin2 (2, ro) sin 0 ...... 2 2 (8)

...... 1-- •

• 2 (IO, I) . 2 (ro, 2) sm ~-~ sm ~~~ 0 2 2