2009;
240 pp;
Hardcover

MSC: Primary 51;

**Print ISBN: 978-0-8218-4375-8
Product Code: MBK/56**

List Price: $57.00

AMS Member Price: $45.60

MAA Member Price: $51.30

**Electronic ISBN: 978-1-4704-1595-2
Product Code: MBK/56.E**

List Price: $53.00

AMS Member Price: $42.40

MAA Member Price: $47.70

#### Supplemental Materials

# Poncelet’s Theorem

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*Leopold Flatto*

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.

#### Readership

Undergraduate and graduate students interested in projective geometry, complex analysis, dynamical systems, and general mathematics.

#### Reviews & Endorsements

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

# Table of Contents

## Poncelet's Theorem

- Cover Cover11 free
- Title page i3 free
- Contents v7 free
- Preface xi13 free
- List of symbols xv17 free
- Introduction 119 free
- Part I. Projective geometry 1331 free
- Basic notions of projective geometry 1533
- Conics 3149
- Intersection of two conics 4361
- Part II. Complex analysis 5977
- Riemann surfaces 6179
- Elliptic functions 83101
- The modular function 97115
- Elliptic curves 111129
- Part III. Poncelet and Cayley theorems 121139
- Poncelet’s theorem 123141
- Cayley’s theorem 135153
- Non-generic cases 141159
- The real case of Poncelet’s theorem 153171
- Part IV. Related topics 163181
- Billiards in an ellipse 165183
- Double queues 179197
- Part V. Supplement 189207
- Billiards and the Poncelet theorem 191209
- Part VI. Appendices 213231
- Factorization of homogeneous polynomials 215233
- Degenerate conics of a conic pencil. Proof of Theorem 4.9 219237
- Lifting theorems 223241
- Proof of Theorem 11.5 229247
- Billiards in an ellipse. Proof of Theorem 13.1 233251
- References 237255
- Index 239257 free
- Back Cover Back Cover1259