Isometric Billiards in Ellipses and Focal Billiards in Ellipsoids

TL;DR

This paper establishes a continuous isometric relationship between elliptical and focal billiards in ellipsoids, enabling transfer of properties and insights between planar and spatial billiard systems.

Contribution

It proves the existence of an isometric correspondence between elliptical and focal billiards, and introduces a method to transfer properties from planar to spatial billiards via focal billiards in ellipsoids.

Findings

Existence of isometric counterparts for elliptical and focal billiards.

Transfer of billiard properties and parametrizations between planar and spatial cases.

Connection between periodic billiards and Poncelet grids in different geometries.

Abstract

Billiards in ellipses have a confocal ellipse or hyperbola as caustic. The goal of this paper is to prove that for each billiard of one type there exists an isometric counterpart of the other type. Isometry means here that the lengths of corresponding sides are equal. The transition between these two isometric billiard can be carried out continuosly via isometric focal billiards in a fixed ellipsoid. The extended sides of these particular billiards in an ellipsoid are focal axes, i.e., generators of confocal hyperboloids. This transition enables to transfer properties of planar billiards to focal billiards, in particular billiard motions and canonical parametrizations. A periodic planar billiard and its associated Poncelet grid give rise to periodic focal billiards and spatial Poncelet grids. If the sides of a focal billiard are materialized as thin rods with spherical joints at the vertices and other crossing points between different sides, then we obtain Henrici's hyperboloid, which is flexible between the two planar limits.