The return map of a Birkhoff billiard is an area-preserving map on the annulus. Birkhof billiards become so a natural object in ergodic theory . In some cases, these sets are Cantor sets, sometimes invariant curves. There exists a maximum and so a critical point. Curves for which tangent billiard trajectories remains tangent after successive reflections.
Ellipses have conformal conics as caustics, hyperbola for , ellipses for .
Curves of equal width have caustics which agree with the evolute of the table. Furthermore , so that . If additionally , this is a table of equal width . This is the evolute of the curve. Vertices of table cusps of .