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arXiv · 2606.04132

On the First Caustic of Elliptical Billiards

Aleksandra Uskova
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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 $\Gamma_n$, 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, $\Gamma_n$ 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 $\Gamma_n$ has at least four ordinary cusps. In this paper, we present a proof that $\Gamma_1$ has exactly four ordinary cusps, using billiards in complex spaces.

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Aleksandra Uskova. 2026-06-02. On the First Caustic of Elliptical Billiards. https://arxiv.org/abs/2606.04132

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