On cusps of caustics by reflection: a billiard variation on Jacobi's Last Geometric Statement

TL;DR

This paper investigates the geometric properties of caustics formed by light reflecting inside an oval, proving that each caustic has at least four cusps using diverse mathematical techniques.

Contribution

It establishes a lower bound on the number of cusps of caustics in a billiard reflection setting, extending Jacobi's classical geometric statement.

Findings

Each caustic has at least 4 cusps for a generic source.

Multiple proofs are provided, including methods from curve shortening flow and Legendrian knot theory.

The results connect billiard dynamics with classical differential geometry.

Abstract

A point source of light is placed inside an oval. The $n$-th caustic by reflection is the envelope of the light rays emanating from the light source after $n$ reflections off the curve. We show that each of these caustics, for a generic point light source, has at least 4 cusps. This is a billiard variation on Jacobi's Last Geometric Statement, concerning the number of cusps of the conjugate locus of a point on a convex surface. We present various proofs, using different ideas, including the curve shortening flow and Legendrian knot theory.