Abstract: The boundary of a planar oval is an ideal mirror, and one has a point source of light inside the oval. Consider the rays of light that have undergone N reflection in the mirror, where N=1,2,... The envelope of this system of rays is the Nth caustic by reflection. I shall explain that, for every N, this caustic has at least four cusps. Similar problems for convex surfaces were considered before: Caratheodory proved that the locus of points conjugated to a given point has at least four cusps, and Jacobi stated, in his "Lectures on dynamics", that this number is exactly four in the case of the ellipsoid (this is known as the "Last Geometric Statement of Jacobi"). Our problem is a billiard version of these problems of differential geometry of surfaces, and it belongs to an endless collections of results stemmed from and motivated by the classical 4-vertex theorem of Mukhopadhyaya.
Alain Albouy, Alain Chenciner, Jacques Laskar
