Séminaire de Géométrie Complexe

Cusps of caustics by reflection in a convex billiard table

by Gil Bor (Math Research Center (CIMAT), Guanajuato, Mexico)

salle 207 bâtiment 1R2 (Institut de Mathématiques de Toulouse)

salle 207 bâtiment 1R2

Institut de Mathématiques de Toulouse


Place a point light source inside a smooth convex billiard table (or mirror). The n-th caustic by reflection is the envelope of light rays after n reflections. Theorem: each of these caustics, for a generic point light source, has at least 4 cusps. Conjecture: there are exactly 4 cusps iff the table is an ellipse. Here are the 2nd, 5th and 8th caustics by reflection in an ellipse, each with 4 cusps (marked by gray disks; the light source is the white disk)   





This is a billiard version of "Jacobi’s Last Geometric Statement", concerning the number of cusps of the conjugate locus of a point on a convex surface, proved so far only in the n=1 case. I will show various proofs, using curve shortening flow and Legendrian knot theory. (Joint work with Serge Tabachnikov, from Penn State, USA).