On the First Caustic of Elliptical Billiards

Abstract

A point source of light is placed inside a billiard with a smooth, convex, closed boundary. For any integer $n$, the $n$-th caustic by reflection, denoted by $Γ_n$, is the envelope of light rays that have undergone $n$ reflections in such a billiard after emanating from the source. It has been conjectured by Gil Bor and Serge Tabachnikov that for an elliptical billiard, $Γ_n$ has exactly four ordinary cusps; this problem is a billiard variation of Jacobi's Last Geometric Statement, which concerns the number of cusps in the conjugate locus of a point on an ellipsoid. Gil Bor, Mark Spivakovsky, and Serge Tabachnikov have proven that $Γ_n$ has at least four ordinary cusps. In this paper, we present a proof that $Γ_1$ has exactly four ordinary cusps, using billiards in complex spaces.