On Cusps of Caustics by Reflection: Billiard Variations on the Four Vertex Theorem and on Jacobi’s Last Geometric Statement

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Abstract

A point source of light is placed inside an oval. The nth caustic by reflection is the envelope of the light rays emanating from the light source after n reflections off the curve. We show that, for a generic point light source, each of these caustics 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.

Original languageEnglish (US)
Pages (from-to)454-467
Number of pages14
JournalAmerican Mathematical Monthly
Volume130
Issue number5
DOIs
StatePublished - 2023

All Science Journal Classification (ASJC) codes

  • General Mathematics

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