DOCSLIB.ORG

13 Circles and Cross Ratio

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
13 Circles and Cross Ratio 13 Circles and Cross Ratio Following up Klein’s Erlangen Program, after defining the group of linear transformations, we study the invariant properites of subsets in the Riemann Sphere under this group. Appolonius’ Circle Recall that a line or a circle on the complex plane, their corresondence in the Riemann Sphere are all circles. We would like to prove that any linear transformation f maps a circle or a line into another circle or line. To prove this, we may separate 4 cases: f maps a line to a line; f maps a line to a circle; f maps a circle to a line; and f maps a circle to a circle. Also, we observe that f is a composition of several simple linear transformations of three type: 1 f1(z)= az, f2(z)= a + b and f3(z)= z because az + b g(z) f(z)= = = g(z)φ(h(z)) (56) cz + d h(z) 1 where g(z) = az + b, h(z) = cz + d and φ(z) = z . Then it suffices to separate 8 cases: fj maps a line to a line; fj maps a line to a circle; fj maps a circle to a line; and fj maps a circle to a circle for j =1, 2, 3. To simplify this proof, we want to write a line or a circle by a single equation, although usually equations for a line and for a circle are are quite different. Let us consider the following single equation: z − z1 = k, (57) z − z2 for some z1, z2 ∈ C and 0 <k< ∞. We call z − z1 z ∈ C = k z − z2 the Appolonius’s circle. Remark Apollonius (262 B.C.-190 B.C.) was a Greek geometer and astronomer. His major mathematical work on the theory of conic sections had a very great influence on the development of mathematics and his famous book Conics introduced the terms parabola, ellipse and hyperbola. Apollonius’ theory of conics was so admired that it was he, rather than Euclid, who in antiquity earned the title the Great Geometer. Let us consider two cases: k = 1 and k =6 1. 88 2 2 • If k = 1, the equation (57) gives |z − z1| = |z − z2| . =⇒ (z − z1)(z − z1)=(z − z2)(z − z2), 2 2 2 2 i.e., |z| − z1z − zz1 + |z1| = |z| − z2z + z2z + |z2| , 2 2 i.e., −z1z − zz1 + |z1| = −z2z + z2z + |z2| , 2 2 i.e., (z2 − z1)z +(z2 − z1)z + |z1| −|z2| =0. Substituting z = x + iy and z = x − iy into the equation, we obtain 2 2 (z2 − z1 + z2 − z1)x +(iz2 − iz1 − iz2 + iz1)y + |z1| −|z2| =0 (58) z−z Since z + z = Re(z) and 2i = Im(z), the above cofficients of x and y are all real numbers. Recall Ax + By + C =0 with A,B,C ∈ R gives a line, the equation (58) gives a line in the real plane and hence in the complex space C = R2. 2 2 2 • If k =6 1, the equation (57) gives |z − z1| = k |z − z2| . 2 =⇒ (z − z1)(z − z1)= k (z − z2)(z − z2), 2 2 2 i.e., (1 − k)|z| +(−z1 + kz2)z +(−z1 + kz2)z + |z1| −|z2| =0, i.e., 2 2 2 2 (1−k)(x +y )+(−z1 +kz2 −z1 +kz2)x+(−iz1 +ikz2 +iz1 −ikz2)y +|z1| −|z2| =0, (59) Recall that an equation a(x2 + y2)+ bx + cy + d =0, with real numbers a =6 0,b,c and d, gives either a circle or the empty set. Geometricall by considering distance, the equation |z − z1| = k|z − z2| must have solutions. As a result, the equation (59) gives a circle in R2 or in C. Invariance of Circles Under Linear Transformations From now, by a circle in CP1, we mean either a line or a circle in C. The following theorem means that any circle in CP1, under a linear transformation, is mapped into again a circle in CP1. 89 Theorem 13.1 Any linear transformation f maps a circle of Cˆ into a circle of Cˆ. Proof: By Remarks below (55), it sufficient to prove this theorem for special linear trans- 1 formations: f(z)= z + b, f(z)= az and f(z)= z . (i) Consider f(z) = z + b, i.e., z 7→ w = f(z) = z + b. It is equivalent to z = w − b. z−z1 k Substituting this into the Applollonia equation (57) z−z2 = , we obtain w − b − z1 = k, w − b − z2 i.e., w − (b + z1) = k w − (b + z2) which is again Applonia equation. Therefore, as we have proved, this equation gives a circle in Cˆ. w (ii) Consider f(z)= az, i.e., z 7→ w = f(z)= az. It is equivalent to z = a . Substituting z−z1 k this into the Applollonia equation (57) z−z2 = , we obtain w a − z1 w = k, − z2 a i.e., w − az1 = k w − az2 which is again Applonia equation. Therefore, as we have proved, this equation gives a circle in Cˆ. 1 1 w (iii) Consider f(z)= z , i.e., z 7→ w = f(z)= z . It is equivalent to z = a . Substituting this into the Applollonia equation (57), we obtain 1 w − z1 1 = k, − z2 w i.e., 1 − z1w = k. 1 − z2w If z1 =6 0 and z2 =6 0, we get 1 w z1 − |z2| 1 = k, − w |z1| Z2 90 which is again Applonia equation. If z1 = 0(⇒ z2 =6 0), we have 1 w 1 = k, − z2 w i.e., 1 1 1 = k =⇒ w − = . |1 − z2w| z2 k|z2| This is a circle in C by considering distance. If z2 = 0(⇒ z1 =6 0), we have 1 − z1 w = k, 1 w i.e., 1 k |1 − z1w| = k =⇒ w − = . z1 |z1| This is a circle in C by considering distance. Cross ratio For any four distinct points z1, z2, z3, z4 ∈ Cˆ, the cross ratio of z1, z2, z3 and z4 is defined by (z1 − z3)(z2 − z4) (z1, z2, z3, z4) := . (z1 − z4)(z2 − z3) z2−z4 z1−z3 z1 z1, z2, z3, z4 z2 z1, z2, z3, z4 Notice: If = ∞, then ( ) := z2−z3 . If = ∞, then ( ) := z1−z4 . If z2−z3 z1−z3 z3 z1, z2, z3, z4 z4 z1, z2, z3, z4 = ∞, then ( ) := z1−z4 . If = ∞, then ( ) := z2−z3 . Proposition 13.2 For any linear transformation f, and for any four distinct points z1, z2, z3, z4 ∈ Cˆ, we have (z1, z2, z3, z4)=(f(z1), f(z2), f(z3), f(z4)). Namely, cross ratio is invariant under linear transformation. Proof: Define a linear transformation (z − z3)(z2 − z4) g(z) := (z, z2, z3, z4)= . (z − z4)(z2 − z3) and (f(z) − f(z3))(f(z2) − f(z4)) h(z) := (f(z), f(z2), f(z3), f(z4)) = . (f(z) − f(z4))(f(z2) − f(z3)) 91 Since g(z2)= h(z2)=1,g(z3)= h(z3) = 0 and g(z4)= h(z4)= ∞, by Theorem 12.2 and its uniqueness, we conclude g = h so that g(z1)= h(z1), i.e., (z1, z2, z3, z4)=(f(z1), f(z2), f(z3), f(z4)). Remark: From the above proof, given any three distinct numbers z2, z3, z4 ∈ Cˆ, the function g(z) := (z, z2, z3, z4) defines a linear transformation satisfying g(z2)=1,g(z3) = 0 and g(z4)= ∞. This is a way to construct linear transformations. Proposition 13.3 For any four distinct points z1, z2, z3, z4 ∈ Cˆ, they lie in a circle (i.e., a stright line or a circle in C) if and only if the cross tratio (z1, z2, z3, z4) is a real number. Proof: Since the image of circle by a linear transformtaiton is still a circle, choose a linear transformation f such that f(z2)=1, f(z3)=0, f(z4)= ∞. Then z1, z2, z3, z4 lie in a circle ⇐⇒ f(z1), f(z2), f(z3), f(z4) lie in the x-axis. ⇐⇒ f(z1) is real ⇐⇒ (f(z1), f(z2), f(z3), f(z4)) is real ⇐⇒ (z1, z2, z3, z4) is real The last equivalence comes from Proposition 13.2. Non-Euclidean geometry In the Elements, Euclid began with a limited number of assumptions (23 definitions, five common notions, and five postulates) and sought to prove all the other results (propositions) in the work. The most famous part of The Elements is the following five postulates: 1. A straight line segment can be drawn joining any two points. 2. Any straight line segment can be extended indefinitely in a straight line. 3. Given any straight line segment, a circle can be drawn having the segment as radius and one endpoint as center. 4. That all right angles are equal to each other. 92 5. If two lines are drawn which intersect a third in such a way that the sum of the inner angles on one side is less than two right angles, then the two lines inevitably must intersect each other on that side if extended far enough (see above picture). This postulate is equivalent to what is known as the parallel postulate, that is, given a traight line L and a point A not in the line, there exists a unique straight line L′ that is parallel to L and contains the point A. It is apparent that the fifth postulate, which can also be called “Euclid’s Fifth Postulate”, is radically different from the first four. Euclid was not satisfied with it, so he tried to avoid using it in The Elements.
Load more

Recommended publications
  • Projective Geometry: a Short Introduction
    Projective Geometry: A Short Introduction Lecture Notes Edmond Boyer Master MOSIG Introduction to Projective Geometry Contents 1 Introduction 2 1.1 Objective . .2 1.2 Historical Background . .3 1.3 Bibliography . .4 2 Projective Spaces 5 2.1 Definitions . .5 2.2 Properties . .8 2.3 The hyperplane at infinity . 12 3 The projective line 13 3.1 Introduction . 13 3.2 Projective transformation of P1 ................... 14 3.3 The cross-ratio . 14 4 The projective plane 17 4.1 Points and lines . 17 4.2 Line at infinity . 18 4.3 Homographies . 19 4.4 Conics . 20 4.5 Affine transformations . 22 4.6 Euclidean transformations . 22 4.7 Particular transformations . 24 4.8 Transformation hierarchy . 25 Grenoble Universities 1 Master MOSIG Introduction to Projective Geometry Chapter 1 Introduction 1.1 Objective The objective of this course is to give basic notions and intuitions on projective geometry. The interest of projective geometry arises in several visual comput- ing domains, in particular computer vision modelling and computer graphics. It provides a mathematical formalism to describe the geometry of cameras and the associated transformations, hence enabling the design of computational ap- proaches that manipulates 2D projections of 3D objects. In that respect, a fundamental aspect is the fact that objects at infinity can be represented and manipulated with projective geometry and this in contrast to the Euclidean geometry. This allows perspective deformations to be represented as projective transformations. Figure 1.1: Example of perspective deformation or 2D projective transforma- tion. Another argument is that Euclidean geometry is sometimes difficult to use in algorithms, with particular cases arising from non-generic situations (e.g.
    [Show full text]
  • Projective Coordinates and Compactification in Elliptic, Parabolic and Hyperbolic 2-D Geometry
    PROJECTIVE COORDINATES AND COMPACTIFICATION IN ELLIPTIC, PARABOLIC AND HYPERBOLIC 2-D GEOMETRY Debapriya Biswas Department of Mathematics, Indian Institute of Technology-Kharagpur, Kharagpur-721302, India. e-mail: d [email protected] Abstract. A result that the upper half plane is not preserved in the hyperbolic case, has implications in physics, geometry and analysis. We discuss in details the introduction of projective coordinates for the EPH cases. We also introduce appropriate compactification for all the three EPH cases, which results in a sphere in the elliptic case, a cylinder in the parabolic case and a crosscap in the hyperbolic case. Key words. EPH Cases, Projective Coordinates, Compactification, M¨obius transformation, Clifford algebra, Lie group. 2010(AMS) Mathematics Subject Classification: Primary 30G35, Secondary 22E46. 1. Introduction Geometry is an essential branch of Mathematics. It deals with the proper- ties of figures in a plane or in space [7]. Perhaps, the most influencial book of all time, is ‘Euclids Elements’, written in around 3000 BC [14]. Since Euclid, geometry usually meant the geometry of Euclidean space of two-dimensions (2-D, Plane geometry) and three-dimensions (3-D, Solid geometry). A close scrutiny of the basis for the traditional Euclidean geometry, had revealed the independence of the parallel axiom from the others and consequently, non- Euclidean geometry was born and in projective geometry, new “points” (i.e., points at infinity and points with complex number coordinates) were intro- duced [1, 3, 9, 26]. In the eighteenth century, under the influence of Steiner, Von Staudt, Chasles and others, projective geometry became one of the chief subjects of mathematical research [7].
    [Show full text]
  • Lie Group and Geometry on the Lie Group SL2(R)
    INDIAN INSTITUTE OF TECHNOLOGY KHARAGPUR Lie group and Geometry on the Lie Group SL2(R) PROJECT REPORT – SEMESTER IV MOUSUMI MALICK 2-YEARS MSc(2011-2012) Guided by –Prof.DEBAPRIYA BISWAS Lie group and Geometry on the Lie Group SL2(R) CERTIFICATE This is to certify that the project entitled “Lie group and Geometry on the Lie group SL2(R)” being submitted by Mousumi Malick Roll no.-10MA40017, Department of Mathematics is a survey of some beautiful results in Lie groups and its geometry and this has been carried out under my supervision. Dr. Debapriya Biswas Department of Mathematics Date- Indian Institute of Technology Khargpur 1 Lie group and Geometry on the Lie Group SL2(R) ACKNOWLEDGEMENT I wish to express my gratitude to Dr. Debapriya Biswas for her help and guidance in preparing this project. Thanks are also due to the other professor of this department for their constant encouragement. Date- place-IIT Kharagpur Mousumi Malick 2 Lie group and Geometry on the Lie Group SL2(R) CONTENTS 1.Introduction ................................................................................................... 4 2.Definition of general linear group: ............................................................... 5 3.Definition of a general Lie group:................................................................... 5 4.Definition of group action: ............................................................................. 5 5. Definition of orbit under a group action: ...................................................... 5 6.1.The general linear
    [Show full text]
  • Concept of Symmetry in Closure Spaces As a Tool for Naturalization of Information
    数理解析研究所講究録 29 第2008巻 2016年 29-36 Concept of Symmetry in Closure Spaces as a Tool for Naturalization of Information Marcin J. Schroeder Akita Intemational University, Akita, Japan [email protected] 1. Introduction To naturalize information, i.e. to make the concept of information a subject of the study of reality independent from subjective judgment requires methodology of its study consistent with that of natural sciences. Section 2 of this paper presents historical argumentation for the fundamental role of the study of symmetries in natural sciences and in the consequence of the claim that naturalization of information requires a methodology of the study of its symmetries. Section 3 includes mathematical preliminaries necessary for the study of symmetries in an arbitrary closure space carried out in Section 4. Section 5 demonstrates the connection of the concept of information and closure spaces. This sets foundations for further studies of symmetries of information. 2. Symmetry and Scientific Methodology Typical account of the origins of the modern theory of symmetry starts with Erlangen Program of Felix Klein published in 1872. [1] This publication was of immense influence. Klein proposed a new paradigm of mathematical study focusing not on its objects, but on their transformations. His mathematical theory ofgeometric symmetry was understood as an investigation of the invariance with respect to transformations of the geometric space (two‐dimensional plane or higher dimensional space). Klein used this very general concept of geometric symmetry for the purpose of a classification of different types of geometries (Euclidean and non‐Euclidean). The fundamental conceptual framework of Kleins Program (as it became function as a paradigm) was based on the scheme of(1) space as a collection of points \rightarrow (2) algebraic structure (group) of its transformations \rightarrow (3) invariants of the transformations, i.e.
    [Show full text]
  • Cassirer and the Structural Turn in Modern Geometry
    JOURNAL FOR THE HISTORY OF ANALYTICAL PHILOSOPHY CASSIRER AND THE STRUCTURAL TURN IN MODERN VOLUME 6, NUMBER 3 GEOMETRY GeorG SCHIEMER EDITOR IN CHIEF KEVIN C. KLEMENt, UnIVERSITY OF MASSACHUSETTS EDITORIAL BOARD The paper investigates Ernst Cassirer’s structuralist account ANNALISA COLIVA, UnIVERSITY OF MODENA AND UC IRVINE of geometrical knowledge developed in his Substanzbegriff und GrEG FROSt-ARNOLD, HOBART AND WILLIAM SMITH COLLEGES Funktionsbegriff (1910). The aim here is twofold. First, to give HENRY JACKMAN, YORK UnIVERSITY a closer study of several developments in projective geometry SANDRA LaPOINte, MCMASTER UnIVERSITY that form the direct background for Cassirer’s philosophical re- CONSUELO PRETI, THE COLLEGE OF NEW JERSEY marks on geometrical concept formation. Specifically, the paper MARCUS ROSSBERG, UnIVERSITY OF CONNECTICUT will survey different attempts to justify the principle of duality ANTHONY SKELTON, WESTERN UnIVERSITY in projective geometry as well as Felix Klein’s generalization of MARK TEXTOR, KING’S COLLEGE LonDON the use of geometrical transformations in his Erlangen program. AUDREY YAP, UnIVERSITY OF VICTORIA The second aim is to analyze the specific character of Cassirer’s RICHARD ZACH, UnIVERSITY OF CALGARY geometrical structuralism formulated in 1910 as well as in sub- sequent writings. As will be argued, his account of modern REVIEW EDITORS geometry is best described as a “methodological structuralism”, SEAN MORRIS, METROPOLITAN STATE UnIVERSITY OF DenVER that is, as a view mainly concerned with the role of structural SANFORD SHIEH, WESLEYAN UnIVERSITY methods in modern mathematical practice. DESIGN DaNIEL HARRIS, HUNTER COLLEGE JHAPONLINE.ORG SPECIAL ISSUE: METHOD, SCIENCe, AND MATHEMATICS: Neo-KANTIANISM AND ANALYTIC PHILOSOPHY © 2018 GeorG SCHIEMER EDITED BY SCOTT EDGAR AND LyDIA PATTON CASSIRER AND THE STRUCTURAL TURN IN In this paper, we aim to further connect these two lines of re- MODERN GEOMETRY search.
    [Show full text]
  • Cycles Cross Ratio: an Invitation
    CYCLES CROSS RATIO: AN INVITATION VLADIMIR V. KISIL Abstract. The paper introduces cycles cross ratio, which extends the classic cross ratio of four points to Lie sphere geometry. Just like the classic counterpart cycles cross ratio is a measure of anharmonicity between spheres with respect to inver- sion. It also provides a Mobius¨ invariant distance between spheres. Many more further properties of cycles cross ratio awaiting their exploration. 1. Introduction We introduce a new invariant in Lie sphere geometry [3, Ch. 3; 6; 16] which is analogous to the cross-ratio of four points in projective [26; 30, § 9.3; 32, App. A] and hyperbolic geometries [2, § 4.4; 27, III.5; 29, § I.5]. To avoid technicalities and stay visual we will work in two dimensions, luckily all main features are already illuminating in this case. Our purpose is not to give an exhausting presentation (in fact, we are hoping it is far from being possible now), but rather to draw an attention to the new invariant and its benefits to the sphere geometry. There are numerous generalisation in higher dimensions and non-Euclidean metrics, see Sect. 5 for a discussion. There is an interactive version of this paper implemented as a Jupyter note- book [25] based on the MoebInv software package [24]. 2. Projective space of cycles Circles are traditional subject of geometrical study and their numerous prop- erties were already widely presented in Euclid’s Elements. However, advances in their analytical presentations is not a distant history. A straightforward parametrisation of a circle equation: (1) x2 + y2 + 2gx + 2fy + c = 0 by a point (g, f, c) in some subset of the three-dimensional Euclidean space R3 was used in [27, Ch.
    [Show full text]
  • The Concept of Manifold, 1850-1950
    CHAPTER 2 The Concept of Manifold, 1850-1950 Erhard Scholz Fachbereich Mathematik, Bergische Universitat Gesamthochschule Wuppertal, Gausstrasse 20, 42097 Wuppertal, Germany 1. Origin of the manifold concept 1.1. n-dimensional systems geometrized In the early 19th century we find diverse steps towards a generalization of geometric lan­ guage to higher dimensions. But they were still of a tentative and often merely metaphori­ cal character. The analytical description of dynamical systems in classical mechanics was a field in which, from hindsight, one would expect a drive towards and a growing awareness of the usefulness of higher dimensional geometrical language.^ But the sources do not, with some minor exceptions, imply such expectations. Although already Lagrange had used the possibility to consider time as a kind of fourth dimension in addition to the three spatial coordinates of a point in his Mecanique analitique (1788) and applied a contact argument to function systems in 5 variables by transfer from the 3-dimensional geometrical case in his Theorie des fonctions analytiques (1797, Section 3.5.25), these early indications were not immediately followed by others. Not before the 1830-s and 1840-s do we find broader attempts to generalize geometri­ cal language and geometrical ideas to higher dimensions: Jacobi (1834), e.g., calculated the volume of n-dimensional spheres and used orthogonal substitutions to diagonalize quadratic forms in n variables, but preferred to avoid explicit geometrical language in his investigations. Cayley's Chapters in the analytical geometry ofn dimensions (1843) did use such explicit geometrical language - but still only in the title, not in the text of the article.
    [Show full text]
  • Geometric Modelling Summer 2018
    Geometric Modelling Summer 2018 Prof. Dr. Hans Hagen http://hci.uni-kl.de/teaching/geometric-modelling-ss2018 Prof. Dr. Hans Hagen Geometric Modelling Summer 20181 Ane Spaces, Elliptic and Hyperbolic Geometry Ane Spaces, Elliptic and Hyperbolic Geometry Prof. Dr. Hans Hagen Geometric Modelling Summer 20182 Ane Spaces, Elliptic and Hyperbolic Geometry Ane Spaces Euclidean Geometry can be formulated by the invariant theories. Question: What is invariant under Euclidean transformations (rotation, translation)? Answer: length of vectors, angles between vectors, volumes spanned by vectors Question: What is invariant under projective transformations? Answer: dimension of subspaces (linea are mapped to lines etc.), cross ratio In this chapter, we will discuss ane spaces and transformations as well as elliptic and hyperbolic geometry. Prof. Dr. Hans Hagen Geometric Modelling Summer 20183 Ane Spaces, Elliptic and Hyperbolic Geometry Ane Spaces Denition: Ane Space A non-empty set A of elements ("points") is called an Ane Space, if there exist a vector space V and a map 7! s.t. every pair (p; q) of points with p; q 2 A are mapped to exactly one vector ~v 2 V s.t. a) For every ~v 2 V and p 2 A there is exactly one point q 2 A with (p; q) 7! ~v. b) If (p; q) 7! ~v and (q; r) 7! w~ , then (p; r) 7! ~v + w~ . V is called the vector space belonging to A. A base of A is given by A : f0; ~v1; ~v2;:::; ~vng, where 0 2 A and f~v1; ~v2;:::; ~vng form a base of V .
    [Show full text]
  • Symmetry-Based Generative Design: a Teaching Experiment
    3B-011.qxd 4/29/2013 12:28 PM Page 303 R. Stouffs, P. Janssen, S. Roudavski, B. Tunçer (eds.), Open Systems: Proceedings of the 18th International Conference on Computer-Aided Architectural Design Research in Asia (CAADRIA 2013), 303–312. © 2013, The Association for Computer-Aided Architectural Design Research in Asia (CAADRIA), Hong Kong, and Center for Advanced Studies in Architecture (CASA), Department of Architecture-NUS, Singapore. SYMMETRY-BASED GENERATIVE DESIGN: A TEACHING EXPERIMENT José Pedro SOUSA and João Pedro XAVIER University of Porto, Porto, Portugal [email protected], [email protected] Abstract. Throughout history, symmetry has been widely explored as a geometric strategy to conceive architectural forms and spaces. Nonetheless, its concept has changed and expanded overtime, and its design exploration does not mean anymore the generation of simple and predictable solutions. By framing in history this idea, the present paper discusses the relevance of exploring symmetry in architectural design today, by means of computa- tional design and fabrication processes. It confirms the emergence of a renewed interest in the topic based on two main ideas: On the one hand, symmetry-based design supports the generation of unique and apparent complex solutions out of simple geometric rules, in a bottom-up fashion. On the other hand, despite this intricacy, it assures modularity in the design components, which can bring benefits at the construction level. As the back- ground for testing and illustrating its theoretical arguments, this paper describes the work produced in the Constructive Geometry course at FAUP. Keywords. Geometry; symmetry; computational design; digital manufac- turing; education.
    [Show full text]
  • Erlangen Program at Large-1: Geometry of Invariants
    Symmetry, Integrability and Geometry: Methods and Applications SIGMA 6 (2010), 076, 45 pages Erlangen Program at Large-1: Geometry of Invariants Vladimir V. KISIL School of Mathematics, University of Leeds, Leeds LS2 9JT, UK E-mail: [email protected] URL: http://www.maths.leeds.ac.uk/∼kisilv/ Received April 20, 2010, in final form September 10, 2010; Published online September 26, 2010 doi:10.3842/SIGMA.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 rˆole 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; M¨obius transformations 2010 Mathematics Subject Classification: 30G35; 22E46; 30F45; 32F45 Contents 1 Introduction 2 1.1 Background and history ................................ 2 1.2 Highlights of obtained results ............................. 3 1.3 The paper outline .................................... 4 2 Elliptic, parabolic and hyperbolic homogeneous spaces 5 2.1 SL2(R) group and Clifford algebras ......................... 5 2.2 Actions of subgroups .................................. 6 2.3 Invariance of cycles ................................... 10 3 Space of cycles 11 3.1 Fillmore–Springer–Cnops construction (FSCc) .................... 11 3.2 First invariants of cycles ................................ 14 4 Joint invariants: orthogonality and inversions 16 4.1 Invariant orthogonality type conditions .......................
    [Show full text]
  • The Erlangen Program Revisited: a Didactic Perspective
    THE ERLANGEN PROGRAM REVISITED: A DIDACTIC PERSPECTIVE CINZIA BONOTTO One of the ideas which dominates a latge patt of contempo­ rary mathematics is the notion of group. In 1898 Poincare attributed an a priori nature to this concept. He said that the general concept of group is pre-existent to our minds, at least potentially; it imposes itself on us not as a form of our sen­ sibility but rather of our understanding p The era of the dominance of the group concept in geome­ try was inaugurated in the second half of the last century, Figure 1: The schematiwtion ojKlein's work by the German mathematician Felix Klein ( 1849-1925) with a document that has passed into history under the natne of the Erlangen Program This work is in fact also explicitly group but also that of geometric properties invariant with cited, as part of the suggested content for the theme Geom­ respect to the action of such a group In this perspective, etry, in the Italian experimental progratns for mathematics in inclusions are reversed, in the sense that to an enlargement the final tluee yeats of Italian Classical and Technical High of the group there conesponds a reduction of the cone­ Schools (Licei, students 16-18 yeats of age): spending invariant geometric properties. Indeed, if Hand G Equations of isometries and similruity transfOrmations are transformation groups and His a subgroup of G, then the Affinities and their equations Invatiant Properties properties which are invariant for G are also invariant for H, while in general the converse does not hold true.
    [Show full text]
  • A Historical Overview of Connections in Geometry A
    A HISTORICAL OVERVIEW OF CONNECTIONS IN GEOMETRY A Thesis by Kamielle Freeman Bachelor of Arts, Wichita State University, 2008 Submitted to the Department of Mathematics and Statistics and the faculty of the Graduate School of Wichita State University in partial fulfillment of the requirements for the degree of Master of Science May 2011 c Copyright 2011 by Kamielle Freeman All Rights Reserved A HISTORICAL OVERVIEW OF CONNECTIONS IN GEOMETRY The following faculty members have examined the final copy of this thesis for form and content, and recommend that it be accepted in partial fulfillment of the requirement for the degree of Master of Science with a major in Mathematics. Phillip E. Parker, Committee Chair Thalia Jeffres, Committee Member Elizabeth Behrman, Committee Member iii DEDICATION To Pandora. This thesis was accomplished for you and in spite of you. iv \Mathematics is very akin to Art; a mathematical theory not only must be rigorous, but it must also satisfy our mind in quest of simplicity, of harmony, of beauty..." |Charles Ehresmann v ACKNOWLEDGEMENTS I would like to thank Esau Freeman and Denise Unruh for making this thesis possible through their loving support, Dr. Phillip E. Parker for his mentoring and excellent mathe- matical instruction, Emma Traore Zohner for help translating, Raja Balakrishnan and Justin Ryan for their helpful conversations and proof-reading, and all of my friends on the 3rd floor of Jabara Hall for their encouraging words and moral support. vi ABSTRACT This thesis is an attempt to untangle/clarify the modern theory of connections in Geometry. Towards this end a historical approach was taken and original as well as secondary sources were used.
    [Show full text]