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

2021/12/15 by Bor, Gil, Tabachnikov, Serge
#53A04 #Differential Geometry (math.DG) #Dynamical Systems (math.DS) #FOS: Mathematics #Symplectic Geometry (math.SG)

paper · doi:10.48550/arxiv.2112.07852

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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