Elliptical billiards and Poncelet’s theorem

Abstract

Two- and three-dimensional billiard systems with elliptical and ellipsoidal boundaries, respectively, are studied. It is known that the trajectory of such a two-dimensional system generates a caustic conic curve. Many properties of the elliptical billiard system in the language of projective geometry can be described. One of these properties is that if a trajectory is closed after p bounces, then all trajectories sharing the same caustic conic are also closed after p bounces. In projective geometry, this is known as Poncelet’s theorem. Many of the two-dimensional results are extended to three dimensions. In particular, it is shown that all straight segments of a trajectory in a three-dimensional ellipsoidal billiard system are always tangent to two caustic quadrics. If a trajectory is closed after p bounces, then all trajectories sharing the same two caustic quadrics are also closed after p bounces. A generalized Poncelet’s theorem in three dimensions is thus established. On the basis of numerical studies, it is conjectured that this generalized Poncelet’s theorem is also valid for an arbitrary finite number field. The implications of the results are discussed and their extension to n dimensions is outlined.