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Hardcover ISBN:  9780821843758 
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Hardcover ISBN:  9780821843758 
Product Code:  MBK/56 
List Price:  $65.00 
MAA Member Price:  $58.50 
AMS Member Price:  $52.00 
eBook ISBN:  9781470415952 
Product Code:  MBK/56.E 
List Price:  $59.00 
MAA Member Price:  $53.10 
AMS Member Price:  $47.20 
Hardcover ISBN:  9780821843758 
eBook ISBN:  9781470415952 
Product Code:  MBK/56.B 
List Price:  $124.00 $94.50 
MAA Member Price:  $111.60 $85.05 
AMS Member Price:  $99.20 $75.60 

Book Details2009; 240 ppMSC: Primary 51
Poncelet's theorem is a famous result in algebraic geometry, dating to the early part of the nineteenth century. It concerns closed polygons inscribed in one conic and circumscribed about another. The theorem is of great depth in that it relates to a large and diverse body of mathematics. There are several proofs of the theorem, none of which is elementary. A particularly attractive feature of the theorem, which is easily understood but difficult to prove, is that it serves as a prism through which one can learn and appreciate a lot of beautiful mathematics.
The author's original research in queuing theory and dynamical systems figures prominently in the book. This book stresses the modern approach to the subject and contains much material not previously available in book form. It also discusses the relation between Poncelet's theorem and some aspects of queueing theory and mathematical billiards.
The proof of Poncelet's theorem presented in this book relates it to the theory of elliptic curves and exploits the fact that such curves are endowed with a group structure. The book also treats the real and degenerate cases of Poncelet's theorem. These cases are interesting in themselves, and their proofs require some other considerations. The real case is handled by employing notions from dynamical systems.
The material in this book should be understandable to anyone who has taken the standard courses in undergraduate mathematics. To achieve this, the author has included in the book preliminary chapters dealing with projective geometry, Riemann surfaces, elliptic functions, and elliptic curves. The book also contains numerous figures illustrating various geometric concepts.
ReadershipUndergraduate and graduate students interested in projective geometry, complex analysis, dynamical systems, and general mathematics.

Table of Contents

Chapters

Chapter 1. Introduction

Part 1. Projective geometry

Chapter 2. Basic notions of projective geometry

Chapter 3. Conics

Chapter 4. Intersection of two conics

Part II. Complex analysis

Chapter 5. Riemann surfaces

Chapter 6. Elliptic functions

Chapter 7. The modular function

Chapter 8. Elliptic curves

Part III. Poncelet and Cayley theorems

Chapter 9. Poncelet’s theorem

Chapter 10. Cayley’s theorem

Chapter 11. Nongeneric cases

Chapter 12. The real case of Poncelet’s theorem

Part IV. Related topics

Chapter 13. Billiards in an ellipse

Chapter 14. Double queues

Supplement

Chapter 15. Billiards and the Poncelet theorem

Appendices

Appendix A. Factorization of homogeneous polynomials

Appendix B. Degenerate conics of a conic pencil. Proof of Theorem 4.9

Appendix C. Lifting theorems

Appendix D. Proof of Theorem 11.5

Appendix E. Billiards in an ellipse. Proof of Theorem 13.1


Additional Material

Reviews

Physically, the book is compact and beautifully produced, and its 235 pages and 15 chapters reveal its enormous mathematical scope. In fact, it may be said that Leopold Flatto has created a mathematical gem...
MAA Reviews


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Poncelet's theorem is a famous result in algebraic geometry, dating to the early part of the nineteenth century. It concerns closed polygons inscribed in one conic and circumscribed about another. The theorem is of great depth in that it relates to a large and diverse body of mathematics. There are several proofs of the theorem, none of which is elementary. A particularly attractive feature of the theorem, which is easily understood but difficult to prove, is that it serves as a prism through which one can learn and appreciate a lot of beautiful mathematics.
The author's original research in queuing theory and dynamical systems figures prominently in the book. This book stresses the modern approach to the subject and contains much material not previously available in book form. It also discusses the relation between Poncelet's theorem and some aspects of queueing theory and mathematical billiards.
The proof of Poncelet's theorem presented in this book relates it to the theory of elliptic curves and exploits the fact that such curves are endowed with a group structure. The book also treats the real and degenerate cases of Poncelet's theorem. These cases are interesting in themselves, and their proofs require some other considerations. The real case is handled by employing notions from dynamical systems.
The material in this book should be understandable to anyone who has taken the standard courses in undergraduate mathematics. To achieve this, the author has included in the book preliminary chapters dealing with projective geometry, Riemann surfaces, elliptic functions, and elliptic curves. The book also contains numerous figures illustrating various geometric concepts.
Undergraduate and graduate students interested in projective geometry, complex analysis, dynamical systems, and general mathematics.

Chapters

Chapter 1. Introduction

Part 1. Projective geometry

Chapter 2. Basic notions of projective geometry

Chapter 3. Conics

Chapter 4. Intersection of two conics

Part II. Complex analysis

Chapter 5. Riemann surfaces

Chapter 6. Elliptic functions

Chapter 7. The modular function

Chapter 8. Elliptic curves

Part III. Poncelet and Cayley theorems

Chapter 9. Poncelet’s theorem

Chapter 10. Cayley’s theorem

Chapter 11. Nongeneric cases

Chapter 12. The real case of Poncelet’s theorem

Part IV. Related topics

Chapter 13. Billiards in an ellipse

Chapter 14. Double queues

Supplement

Chapter 15. Billiards and the Poncelet theorem

Appendices

Appendix A. Factorization of homogeneous polynomials

Appendix B. Degenerate conics of a conic pencil. Proof of Theorem 4.9

Appendix C. Lifting theorems

Appendix D. Proof of Theorem 11.5

Appendix E. Billiards in an ellipse. Proof of Theorem 13.1

Physically, the book is compact and beautifully produced, and its 235 pages and 15 chapters reveal its enormous mathematical scope. In fact, it may be said that Leopold Flatto has created a mathematical gem...
MAA Reviews