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Geometry and Billiards
Serge Tabachnikov, Penn State, University Park, PA

Student Mathematical Library
2005; 176 pp; softcover
Volume: 30
ISBN-10: 0-8218-3919-5
ISBN-13: 978-0-8218-3919-5
List Price: US$39
Member Price: US$31.20
Order Code: STML/30
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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.


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


"(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

  • Motivation: Mechanics and optics
  • Billiard in the circle and the square
  • Billiard ball map and integral geometry
  • Billiards inside conics and quadrics
  • Existence and non-existence of caustics
  • Periodic trajectories
  • Billiards in polygons
  • Chaotic billiards
  • Dual billiards
  • Bibliography
  • Index
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