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

Serge Tabachnikov

Department of Mathematics, Pennsylvania State University, PA, USA

Abstract: Given a projective Finsler metric in a convex domain in the projective plane, that is, a metric in which geodesics are straight lines, consider the respective Finsler billiard system. Choose a generic point inside the table and consider the billiard trajectories that start at this point and undergo $N$ reflection off the boundary. The envelope of the resulting 1-parameter family of straight lines is the $N$th caustic by reflection. We prove that, for every $N$, it has at least four cusps, generalizing a similar result for Euclidean metric, obtained recently jointly with G. Bor.

Keywords: caustic, Finsler billiards, projective Finsler metrics.

MSC: Primary 78A05; Secondary 37C83, 53A04

Received: 09.01.2025

Language: English

DOI: 10.2298/TAM250109004T