On cusps of caustics by reflection in two dimensional projective Finsler metrics

TL;DR

This paper investigates caustics in billiards within convex domains equipped with projective Finsler metrics, proving that such caustics must have at least four cusps, extending classical Euclidean billiard results.

Contribution

It establishes a lower bound of four cusps for caustics in Finsler billiards, generalizing known Euclidean billiard properties to the Finsler setting.

Findings

Caustics in Finsler billiards have at least four cusps.

Extension of Euclidean billiard results to Finsler metrics.

Connection to the geometric properties of conjugate loci in ellipsoids.

Abstract

A Finsler, not necessarily symmetric, metric in the plane or its convex subset is called projective if its geodesics are straight segments. We consider Finsler billiards in a convex planar domain endowed with a projective Finsler metric. A caustic by reflection is the envelope of the oriented lines, the billiard trajectories, that start at a point inside the billiard and undergo a fixed number of reflections. We show that such a caustic has at least four cusps. This problem is motivated by the "Last Geometric Statement of Jacobi" that the conjugate locus of a non-umbilic point of a triaxial ellipsoid has exactly four cusps. The present note extends the recent results in this direction concerning Euclidean billiards.