A Correspondence between Billiards and Geodesics

TL;DR

This paper explores the deep geometric relationship between billiard trajectories and geodesics, establishing a correspondence that allows approximation of billiard paths by geodesic segments on certain surfaces, including non-Euclidean examples.

Contribution

It proves the existence of fold-type surfaces where geodesic segments approximate billiard trajectories, extending known results to new classes of Riemannian tables.

Findings

Existence of fold-type surfaces for approximation

Approximation of billiard trajectories by geodesics in convex Euclidean tables

Extension to non-Euclidean billiard tables

Abstract

From a geometric viewpoint, billiard trajectories and geodesics are related by mutual approximation results. In one direction, it is known that every geodesic curve in the boundary of a smooth convex body can be approximated by a sequence of billiard trajectories inside of it. We establish the other direction by proving that, for Riemannian billiard tables (under mild assumptions), there exists a family of fold-type surfaces such that every sequence of geodesic segments on these surfaces has a subsequence that converges to a billiard trajectory in the table. In particular, this is true for convex Euclidean tables. We also describe a more general class of tables to which this result applies and present explicit non-Euclidean examples.