Nowhere differentiable caustics on spheres

The caustic of a point on a Riemannian manifold is defined as the set of intersection points of infinitesimally closed geodesics at this point. A motivating picture for studying billiards is that the billiard flow can be considered as a limiting case of the geodesic flow. One can now ask, whether caustics of billiards can be related to caustics of a family of geodesics starting at a point in a two-dimensional Riemannian manifold M. Here, in the case of billiards on tables of constant width, an orbit tangent to the caustic can actually be extended to an orbit of a geodesic flow on a sphere.