Open Problems on Billiards and Geometric Optics

Elemente der Mathematik, 1998

We illustrate the dynamics of billiards in tables of constant width. Caustics of such tables can be nowhere differentiable and are related to caustics appearing in differential geometry.

Proceedings of Symposia in Pure Mathematics, 2006

2017

In this paper we shall describe recent applications of billiards in aerodynamics and optics. More precisely, we shall explain how to construct perfectly streamlining bodies in the framework of Newtonian aerodynamics and invisible objects in geometric optics. The methods we shall use are quite elementary and accessible to students of the high school; they include focal properties of curves of the second order and unfolding of a billiard trajectory.

European Journal of Mathematics

Inverting the vertices of elliptic billiard N-periodics with respect to a circle centered on one focus yields a new "focus-inversive" family inscribed in Pascal's Limaçon. The following are some of its surprising invariants: (i) perimeter, (ii) sum of cosines, and (iii) sum of distances from inversion center (the focus) to vertices. We prove these for the N=3 case, showing that this family (a) has a stationary Gergonne point, (b) is a 3-periodic family of a second, rigidly moving elliptic billiard, and (c) the loci of incenter, barycenter, circumcenter, orthocenter, nine-point center, and a great many other triangle centers are circles.

2008

We identify all translation covers among triangular billiard surfaces. Our main tools are the holonomy field of Kenyon and Smillie and a geometric property of triangular billiard surfaces, which we call the fingerprint of a point, that is preserved under balanced translation covers.

Given a strictly convex domain Ω ⊂ R 2 , there is a natural way to define a billiard map in it: a rectilinear path hitting the boundary reflects so that the angle of reflection is equal to the angle of incidence. In this paper we answer a relatively old question of Guillemin. We show that if two billiard maps are C 1,α-conjugate near the boundary, for some α > 1/2, then the corresponding domains are similar, i.e. they can be obtained one from the other by a rescaling and an isometry. As an application, we prove a conditional version of Birkhoff conjecture on the integrability of planar billiards and show that the original conjecture is equivalent to what we call an Extension problem. Quite interestingly, our result and a positive solution to this extension problem would provide an answer to a closely related question in spectral theory: if the marked length spectra of two domains are the same, is it true that they are isometric?

Nonlinearity, 2011

The question of invisibility for bodies with mirror surface is studied in the framework of geometrical optics. We construct bodies that are invisible/have zero resistance in two mutually orthogonal directions, and prove that there do not exist bodies which are invisible/have zero resistance in all possible directions of incidence.

2003

Let M∈ R be a convex Euclidean polyhedron. A generalized diagonal (g.d., for brevity) is said to be a billiard trajectory inside M that starts at some vertex A ∈M and ends at some other (perhaps the same) vertex B ∈M reflecting from interior points of M’s (d − 1)-dimensional faces (see [1]). Note that a g.d. is not actually a real billiard trajectory, because a billiard trajectory must reflect from interior points of a polyhedron’s faces of codimension 1. However, except for both of its ends, the g.d. can be thought of as a piece of a billiard trajectory, meaning that all of its remaining reflection points do not belong to a polyhedron face of dimension < d− 1. We consider the following two special generalized diagonals, Γ and γ, inside M. Let Π be the “horizontal” (d − 1)-dimensional face of polyhedron M (so the whole polyhedron is entirely located on the upper half-space of R), let Γ = A1B1A2B2 . . . . . . An−1Bn−1An be the first, “long”, generalized diagonal, and let γ = A1BnA...

We consider a one-parameter family of billiard tables T ℓ which have as a common caustic the equilateral triangle γ. The billiard tables T ℓ are constructed geometrically by the string construction, where the length ℓ of the string is the parameter. We study the family of circle homeomorphisms f ℓ obtained by restricting the billiard map to the canonical invariant circle Γ ℓ belonging to the caustic and the rotation function ρ(ℓ) = ρ(f ℓ). We show that the graph of ρ is a devil's staircase. We analyze the passage of a Birkhoff periodic orbits through the caustic as the parameter changes.

Journal of Geometry and Physics, 2016

We consider the following problem: given two parallel and identically oriented bundles of light rays in R n+1 and given a diffeomorphism between the rays of the former bundle and the rays of the latter one, is it possible to realize this diffeomorphism by means of several mirror reflections? We prove that a 2-mirror realization is possible, if and only if the diffeomorphism is the gradient of a function. We further prove that any orientation reversing diffeomorphism of domains in R 2 is locally the composition of two gradient diffeomorphisms, and therefore can be realized by 4 mirror reflections of light rays in R 3 , while an orientation preserving diffeomorphism can be realized by 6 reflections. In general, we prove that an (orientation reversing or preserving) diffeomorphism of wave fronts of two normal families of light rays in R 3 can be realized by 6 or 7 reflections.

Communications in Mathematical Physics, 2006

We investigate the rotation sets of billiards on the m-dimensional torus with one small convex obstacle and in the square with one small convex obstacle. In the first case the displacement function, whose averages we consider, measures the change of the position of a point in the universal covering of the torus (that is, in the Euclidean space), in the second case it measures the rotation around the obstacle. A substantial part of the rotation set has usual strong properties of rotation sets.

Journal of Dynamical and Control Systems, 2019

We show that the maximum number of directions of invisibility in a planar billiard defined in the exterior of a piecewise smooth body is at most finite.

Nonlinearity, 2002

We study an area preserving map of the exterior of a smooth convex curve in the hyperbolic plane, defined by a natural geometrical construction and called the dual billiard map. We consider two problems: stability and the area spectrum. The dual billiard map is called stable if all its orbits are bounded. We show that both stable and unstable behaviours may occur. If the map at infinity has a hyperbolic periodic orbit, then the dual billiard map has orbits escaping to infinity. On the other extreme, if the map at infinity is smoothly conjugated to a Diophantine irrational rotation of the circle, then the dual billiard map is stable. The area spectrum is the set of extremal areas of n-gons, circumscribed about the dual billiard curve; this is to the dual billiard what the length spectrum is to the usual, inner, one. We show that the area spectrum has an asymptotic expansion in even negative powers of n as n → ∞. The first coefficient of this expansion is the area of the dual billiard curve, and the next is, up to a constant, the cubed integral of the cube root of its curvature. We describe the curves that are relative extrema of these two functionals and show that they are the trajectories of the pseudospherical pendulum with various gravity directions.

Illinois Journal of Mathematics, 2014

We identify all translation covers among triangular billiard surfaces. Our main tools are the holonomy field of Kenyon and Smillie and a geometric property of triangular billiard surfaces, which we call the fingerprint of a point, that is preserved under balanced translation covers.

Journal of Mathematical Analysis and Applications, 1997

The purpose of this paper is to show that for a dense G set of three smooth