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List of Commonly Used Symbols xv Chapter 1. Introduction 1 1.1. The Theorems of Poncelet and Cayley 1 1.2. The Poncelet and Steiner Theorems- A Misleading Analogy 6 1.3. The Real Case of Poncelet's Theorem 9 1.4. Related Topics 10 Part I. Projective Geometry Chapter 2. Basic Notions of Projective Geometry 15 2.1. Projective Plane 15 2.2. Projectivities 19 2.3. Projective Line 22 2.4. Algebraic Curves 24 Chapter 3. Conics 31 3.1. Conics 31 3.2. Intersection of Line and Conic 34 3.3. Reduced Form 36 3.4. Projective Structure on a Smooth Conic 38 3.5. Parametric Equations of Smooth Conics 39 Chapter 4. Intersection of Two Conics 43 4.1. Intersection Numbers 43 4.2. Bezout's Theorem for Conics 51 4.3. Conic Pencils 53 4.4. Degenerate Conics in a Conic Pencil 55 Part II. Complex Analysis Chapter 5. Riemann Surfaces 61 5.1. Definition of Riemann Surface 61 5.2. Examples of Riemann Surfaces 65 5.3. More Examples of Riemann Surfaces. Algebraic Curves 68 5.4. Examples of Conformal Maps 74 5.5. Covering Surfaces 76 5.6. Isomorphisms of Tori 79 Chapter 6. Elliptic Functions 83 6.1. Elliptic Functions 83 6.2. The Weierstrass p-Function 86 6.3. The Functions ( and a 89 6.4. Differential Equation for p 92 6.5. The Elliptic Function w = sn(z) 94 Chapter 7. The Modular Function 97 7.1. The Functions 92, 93 97 7.2. The Modular Function J 98 7.3. Fundamental Region for r 100 7.4. Fourier Expansion of J 102 7.5. Values of J 104 7.6. Solution to the Inversion Problem 108 Chapter 8. Elliptic Curves 111 8.1. Elliptic Curves 111 8.2. Algebraic Models 113 8.3. Division Points of C/A 115 8.4. Division Points of S 117 Part III. Poncelet and Cayley Theorems Chapter 9. Poncelet's Theorem 123 9.1. Poncelet Correspondence 123 9.2. Algebraic Equation for M 125 9.3. Complex Structure on M 128 9.4. M is an Elliptic Curve 130 9.5. The Automorphisms a, r, and 7 131 9.6. Proof of Poncelet's Theorem 132 Chapter 10. Cayley's Theorem 135 10.1. Origin of MA 135 10.2. Algebraic Equation for M 136 10.3. Proof of Cayley's Theorem 138 Chapter 11. Non-generic Cases 141 11.1. Fixed Points of rl 141 11.2. Equations for C, D, and M 142 11.3. The Riemann Surface AMo 144 11.4. Formulas for i7 147 11.5. Poncelet's Theorem 148 11.6. Existence of Circuminscribed n-Gons 150 Chapter 12. The Real Case of Poncelet's Theorem 153 12.1. Poncelet's Theorem for Two Circles 153 12.2. Poncelet's Theorem for Two Ellipses 155 12.3. Topological Conjugacy 157 Part IV. Related Topics Chapter 13. Billiards in an Ellipse 165 13.1. Billiards in an Ellipse. Caustics 165 13.2. The Map rJR 167 13.3. Description of MR 168 13.4. Invariant Measure. Rotation Number 170 13.5. Billiard Trajectories with the Same Caustic 172 13.6. Derivation of Invariant Measure 173 13.7. Proofs of Theorems 13.3 and 13.4 177 Chapter 14. Double Queues 179 14.1. The Two-Demands Model 180 14.2. Formulas 182 14.3. Riemann Surface 183 14.4. Automorphy Conditions 184 14.5. The Regions Dz and D,, 184 14.6. Analytic Continuation 186 Supplement Chapter 15. Billiards and the Poncelet Theorem S. TABACHNIKOV 191 15.1. Mathematical Billiards 191 15.2. Integrable Case 195 15.3. Poncelet Grid 198 15.4. Poncelet Theorem on Quadratic Surfaces 204 15.5. Outer Billiards in the Hyperbolic Plane 207 References 210 Appendices Appendix A. Factorization of Homogeneous Polynomials 215 Appendix B. Degenerate Conics of a Conic Pencil. Proof of Theorem 4.9 219 Appendix C. Lifting Theorems 223 C.1. Homotopy 223 C.2. Lifting Theorems 224 Appendix D. Proof of Theorem 11.5 229 Appendix E. Billiards in an Ellipse. Proof of Theorem 13.1 233