tailieunhanh - Geometry and Billiards

The applications of mathematics. The usual reasons given in school mathematics for studying mathematics are because it is beautiful, for “men- tal discipline,” or a subject needed by an educated person. These reasons are naive. It doesn’t matter if students find the subject beautiful or even like it. Doing mathematics isn’t like reading Shakespeare, something that every educated person should do, but that seldom has direct relevance to an adult’s everyday life in our society. The main reason for studying math- ematics is that our society could not even function without the applications of a very high level of mathematical knowledge. Consequently, without a real understanding of mathematics one. | Geometry and Billiards Serge Tabachnikov Department of Mathematics Penn State UNivERsiTy Park pa 16802 1991 Mathematics Subject Classification. Primary 37-02 51-02 Secondary 49-02 70-02 78-02 Contents Foreword MASS and REU at Penn State University vii Preface ix Chapter 1. Motivation Mechanics and Optics 1 Chapter 2. Billiard in the Circle and the Square 21 Chapter 3. Billiard Ball Map and Integral Geometry 33 Chapter 4. Billiards inside Conics and Quadrics 51 Chapter 5. Existence and Non-existence of Caustics 73 Chapter 6. Periodic Trajectories 99 Chapter 7. Billiards in Polygons 113 Chapter 8. Chaotic Billiards 135 Chapter 9. Dual Billiards 147 Bibliography 167

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