Geometry And Billiards
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| Author | : Serge Tabachnikov |
| Publisher | : American Mathematical Soc. |
| Total Pages | : 192 |
| Release | : 2005 |
| Genre | : Mathematics |
| ISBN | : 0821839195 |
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 Poincare 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 volume has been copublished with the Mathematics Advanced Study Semesters program at Penn State.
| Author | : Nikolai Chernov |
| Publisher | : American Mathematical Soc. |
| Total Pages | : 330 |
| Release | : 2006 |
| Genre | : Mathematics |
| ISBN | : 0821840967 |
This book covers one of the most exciting but most difficult topics in the modern theory of dynamical systems: chaotic billiards. In physics, billiard models describe various mechanical processes, molecular dynamics, and optical phenomena. The theory of chaotic billiards has made remarkable progress in the past thirty-five years, but it remains notoriously difficult for the beginner, with main results scattered in hardly accessible research articles. This is the first and so faronly book that covers all the fundamental facts about chaotic billiards in a complete and systematic manner. The book contains all the necessary definitions, full proofs of all the main theorems, and many examples and illustrations that help the reader to understand the material. Hundreds of carefullydesigned exercises allow the reader not only to become familiar with chaotic billiards but to master the subject. The book addresses graduate students and young researchers in physics and mathematics. Prerequisites include standard graduate courses in measure theory, probability, Riemannian geometry, topology, and complex analysis. Some of this material is summarized in the appendices to the book.
| Author | : Utkir A Rozikov |
| Publisher | : World Scientific |
| Total Pages | : 223 |
| Release | : 2018-12-06 |
| Genre | : Science |
| ISBN | : 9813276487 |
'This book offers one of the few places where a collection of results from the literature can be found … The book has an extensive bibliography … It is very nice to have the compendium of results that is presented here.'zbMATHA mathematical billiard is a mechanical system consisting of a billiard ball on a table of any form (which can be planar or even a multidimensional domain) but without billiard pockets. The ball moves and its trajectory is defined by the ball's initial position and its initial speed vector. The ball's reflections from the boundary of the table are assumed to have the property that the reflection and incidence angles are the same. This book comprehensively presents known results on the behavior of a trajectory of a billiard ball on a planar table (having one of the following forms: circle, ellipse, triangle, rectangle, polygon and some general convex domains). It provides a systematic review of the theory of dynamical systems, with a concise presentation of billiards in elementary mathematics and simple billiards related to geometry and physics.The description of these trajectories leads to the solution of various questions in mathematics and mechanics: problems related to liquid transfusion, lighting of mirror rooms, crushing of stones in a kidney, collisions of gas particles, etc. The analysis of billiard trajectories can involve methods of geometry, dynamical systems, and ergodic theory, as well as methods of theoretical physics and mechanics, which has applications in the fields of biology, mathematics, medicine, and physics.
| Author | : Vladimir Dragović |
| Publisher | : Springer Science & Business Media |
| Total Pages | : 293 |
| Release | : 2011-05-02 |
| Genre | : Mathematics |
| ISBN | : 3034800150 |
The goal of the book is to present, in a complete and comprehensive way, areas of current research interlacing around the Poncelet porism: dynamics of integrable billiards, algebraic geometry of hyperelliptic Jacobians, and classical projective geometry of pencils of quadrics. The most important results and ideas, classical as well as modern, connected to the Poncelet theorem are presented, together with a historical overview analyzing the classical ideas and their natural generalizations. Special attention is paid to the realization of the Griffiths and Harris programme about Poncelet-type problems and addition theorems. This programme, formulated three decades ago, is aimed to understanding the higher-dimensional analogues of Poncelet problems and the realization of the synthetic approach of higher genus addition theorems.
| Author | : Serge Tabachnikov |
| Publisher | : SMF |
| Total Pages | : 142 |
| Release | : 1995 |
| Genre | : Billiards |
| ISBN | : 9782856290309 |
| Author | : A.V. Bolsinov |
| Publisher | : CRC Press |
| Total Pages | : 747 |
| Release | : 2004-02-25 |
| Genre | : Mathematics |
| ISBN | : 0203643429 |
Integrable Hamiltonian systems have been of growing interest over the past 30 years and represent one of the most intriguing and mysterious classes of dynamical systems. This book explores the topology of integrable systems and the general theory underlying their qualitative properties, singularites, and topological invariants. The authors,
| Author | : Richard Evan Schwartz |
| Publisher | : American Mathematical Soc. |
| Total Pages | : 330 |
| Release | : 2011 |
| Genre | : Mathematics |
| ISBN | : 0821853686 |
The goal of the book is to present a tapestry of ideas from various areas of mathematics in a clear and rigorous yet informal and friendly way. Prerequisites include undergraduate courses in real analysis and in linear algebra, and some knowledge of complex analysis. --from publisher description.
| Author | : Tim Ander |
| Publisher | : CRB Publishing |
| Total Pages | : 42 |
| Release | : 2018-03-10 |
| Genre | : Sports & Recreation |
| ISBN | : 8827502343 |
Take Your Pool Skills to the Next Level and Win Big! Inside How to Play Pool, you’ll discover the rules for many popular variations of the game: Eight-Ball Nine-Ball One-Pocket and Snooker With this book, you can strengthen your pool game with the right posture, physics, and geometry. You’ll learn to execute many different types of shots, such as straight, angled, and spin shots. For example, you’ll learn to combine top/back with left/right spin and get all kinds of impressive results! How to Play Pool explains how you can use your cunning to plan ahead and out-strategize your opponents. You’ll find out why to use just the right amount of force to avoid reflections and “own” pockets. By targeting clumps of balls, you can set yourself up for a great endgame layout. If you pay close attention to the cue ball’s trajectory after it hits the target ball, you’ll set yourself up for shot after easy shot. With these simple and powerful pool-playing tips and techniques, you’ll dominate the table – and the competition! You’ll even learn how to pull off a variety of crowd-pleasing trick shots: Pocketing the Eight-Ball on the Break Jumping Over Obstacles Sinking the 4-in-a-Line Shot Don’t wait – Take the plunge and become a pool shark today with How to Play Pool! It’s fast and easy to order – just scroll up and click the BUY NOW WITH ONE CLICK button on the right-hand side of your screen.
| Author | : Arthur "Babe" Cranfield |
| Publisher | : Simon and Schuster |
| Total Pages | : 300 |
| Release | : 2016-11-01 |
| Genre | : Sports & Recreation |
| ISBN | : 1510712313 |
Learn tips and tactics from the very best, in this newly revised and expanded edition. * Consumer reviews say it best: pool Hall of Famer Arthur “Babe” Cranfield wrote an "easy to read and understand" pool manual that will have "beginners and skilled players alike" play better. "Excellent guide", "helpful illustrations", "recommended to all". * Give it a try and "you cannot help but play better".
| Author | : Richard Evan Schwartz |
| Publisher | : Princeton University Press |
| Total Pages | : 280 |
| Release | : 2019-02-19 |
| Genre | : Mathematics |
| ISBN | : 0691181381 |
Outer billiards provides a toy model for planetary motion and exhibits intricate and mysterious behavior even for seemingly simple examples. It is a dynamical system in which a particle in the plane moves around the outside of a convex shape according to a scheme that is reminiscent of ordinary billiards. The Plaid Model, which is a self-contained sequel to Richard Schwartz’s Outer Billiards on Kites, provides a combinatorial model for orbits of outer billiards on kites. Schwartz relates these orbits to such topics as polytope exchange transformations, renormalization, continued fractions, corner percolation, and the Truchet tile system. The combinatorial model, called “the plaid model,” has a self-similar structure that blends geometry and elementary number theory. The results were discovered through computer experimentation and it seems that the conclusions would be extremely difficult to reach through traditional mathematics. The book includes an extensive computer program that allows readers to explore the materials interactively and each theorem is accompanied by a computer demonstration.
