**Student Mathematical Library**

Volume: 30;
2005;
176 pp;
Softcover

MSC: Primary 37; 51;
Secondary 49; 70; 78

**Print ISBN: 978-0-8218-3919-5
Product Code: STML/30**

List Price: $44.00

Individual Price: $35.20

**Electronic ISBN: 978-1-4704-2141-0
Product Code: STML/30.E**

List Price: $41.00

Individual Price: $32.80

#### Supplemental Materials

# Geometry and Billiards

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*Serge Tabachnikov*

Mathematical billiards describe the motion of a mass point in
a domain with elastic reflections off the boundary or, equivalently,
the behavior of rays of light in a domain with ideally reflecting
boundary. From the point of view of differential geometry, the
billiard flow is the geodesic flow on a manifold with boundary. This
book is devoted to billiards in their relation with differential
geometry, classical mechanics, and geometrical optics.

Topics covered include variational principles of billiard motion,
symplectic geometry of rays of light and integral geometry, existence
and nonexistence of caustics, optical properties of conics and
quadrics and completely integrable billiards, periodic billiard
trajectories, polygonal billiards, mechanisms of chaos in billiard
dynamics, and the lesser-known subject of dual (or outer)
billiards.

The book is based on an advanced undergraduate topics
course. Minimum prerequisites are the standard material covered in the
first two years of college mathematics (the entire calculus sequence,
linear algebra). However, readers should show some mathematical
maturity and rely on their mathematical common sense.

A unique feature of the book is the coverage of many diverse topics
related to billiards, for example, evolutes and involutes of plane
curves, the four-vertex theorem, a mathematical theory of rainbows,
distribution of first digits in various sequences, Morse theory, the
Poincaré recurrence theorem, Hilbert's fourth problem, Poncelet
porism, and many others. There are approximately 100
illustrations.

The book is suitable for advanced undergraduates, graduate
students, and researchers interested in ergodic theory and
geometry.

This book is published in cooperation with Mathematics Advanced Study Semesters

#### Readership

Advanced undergraduates, graduate students, and research mathematicians interested in ergodic theory and geometry.

#### Reviews & Endorsements

(This book) is very well written, with nice illustrations. The author presents the results very clearly, with interesting digressions and he mentions applications of billiards to various fields.

-- Zentralblatt MATH

#### Table of Contents

# Table of Contents

## Geometry and Billiards

- Cover Cover11 free
- Title iii4 free
- Copyright iv5 free
- Contents v6 free
- Foreword: MASS and REU at Penn State University vii8 free
- Preface ix10 free
- Chapter 1. Motivation: Mechanics and Optics 114 free
- Chapter 2. Billiard in the Circle and the Square 2134
- Chapter 3. Billiard Ball Map and Integral Geometry 3346
- Chapter 4. Billiards inside Conies and Quadrics 5164
- Chapter 5. Existence and Non-existence of Caustics 7386
- Chapter 6. Periodic Trajectories 99112
- Chapter 7. Billiards in Polygons 113126
- Chapter 8. Chaotic Billiards 135148
- Chapter 9. Dual Billiards 147160
- Bibliography 167180
- Index 175188
- Back Cover Back Cover1192