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arXiv · 2112.07852

On cusps of caustics by reflection: a billiard variation on Jacobi's Last Geometric Statement

Gil Bor·Serge Tabachnikov
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Abstract

A point source of light is placed inside an oval. The $n$-th caustic by reflection is the envelope of the light rays emanating from the light source after $n$ reflections off the curve. We show that each of these caustics, for a generic point light source, has at least 4 cusps. This is a billiard variation on Jacobi's Last Geometric Statement, concerning the number of cusps of the conjugate locus of a point on a convex surface. We present various proofs, using different ideas, including the curve shortening flow and Legendrian knot theory.

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BibTeXRIS

Gil Bor, Serge Tabachnikov. 2021-12-15. On cusps of caustics by reflection: a billiard variation on Jacobi's Last Geometric Statement. https://arxiv.org/abs/2112.07852

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