By Serge Tabachnikov

Mathematical billiards describe the movement of a mass element in a website with elastic reflections off the boundary or, equivalently, the habit of rays of sunshine in a site with preferably reflecting boundary. From the viewpoint of differential geometry, the billiard circulate is the geodesic movement on a manifold with boundary. This e-book is dedicated to billiards of their relation with differential geometry, classical mechanics, and geometrical optics. subject matters lined contain variational rules of billiard movement, symplectic geometry of rays of sunshine and essential geometry, lifestyles and nonexistence of caustics, optical homes of conics and quadrics and fully integrable billiards, periodic billiard trajectories, polygonal billiards, mechanisms of chaos in billiard dynamics, and the lesser-known topic of twin (or outer) billiards. The publication is predicated on a complicated undergraduate issues path. minimal necessities are the traditional fabric coated within the first years of faculty arithmetic (the complete calculus series, linear algebra). although, readers should still express a few mathematical adulthood and depend on their mathematical logic. a special characteristic of the e-book is the assurance of many different subject matters on the topic of billiards, for instance, evolutes and involutes of aircraft curves, the four-vertex theorem, a mathematical thought of rainbows, distribution of first digits in numerous sequences, Morse conception, the Poincaré recurrence theorem, Hilbert's fourth challenge, Poncelet porism, and so forth. There are nearly a hundred illustrations. The booklet is acceptable for complicated undergraduates, graduate scholars, and researchers drawn to ergodic concept and geometry. This quantity has been copublished with the maths complicated learn Semesters software at Penn nation.

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**Sample text**

An oriented line can be characterized by its direction, an angle ϕ, and its signed distance p from the origin O (the sign of p is that of the frame that consists of the orthogonal vector from the origin to the line and the direction vector of the line). Thus N is a cylinder with coordinates (ϕ, p). 3. Describe the space of non-oriented lines in the plane. 4. Let O′ = O + (a, b) be a different choice of the origin. 1) ϕ′ = ϕ, p′ = p − a sin ϕ + b cos ϕ. The space of lines N has an area form Ω = dϕ ∧ dp.

For a smooth curve γ : [a, b] → M , its Finsler length is given by b L(γ) = L(γ(t), γ ′ (t)) dt. a Due to homogeneity of L, this integral does not depend on the parameterization. 19. Compute the Lagrangian functions for the projective metrics of positive and negative constant curvatures in the plane. 3). Let f (p, ϕ) be a positive continuous function on the space of oriented lines, even with respect to the orientation reversion of a line: f (−p, ϕ + π) = f (p, ϕ). Then one has a new area form: Ωf = f (p, ϕ) dϕ ∧ dp.

3. Square grid partitioned into ladders orbit T i (0), i = 0, . . , n. Since T is an irrational rotation, all these points are distinct and there are n + 1 of them. To describe the initial n-segments of the cutting sequences, start with the line through the origin (0, 0) and parallel translate it along the diagonal of the unit square toward point (−1, 1). The n-segments of the cutting sequence change when the line passes through a vertex of one of the first n ladders. As we have seen, there are n + 1 such events, and hence p(n) = n + 1.