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 Γn, 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, Γn 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 Γn has at least four ordinary cusps. In this paper, we present a proof that Γ1 has exactly four ordinary cusps, using billiards in complex spaces.
Related
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BibTeX
@misc{uskova2026first,
title = {{On the First Caustic of Elliptical Billiards}},
author = {Aleksandra Uskova},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2606.04132},
eprint = {2606.04132},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2606.04132},
}
RIS
TY - UNPB
TI - On the First Caustic of Elliptical Billiards
AU - Aleksandra Uskova
PY - 2026
DA - 2026/11/05
PB - arXiv
DO - 10.48550/arxiv.2606.04132
UR - https://arxiv.org/abs/2606.04132
AB - 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.
ER -