Convex Curves with Periodic Billiard Polygons

Abstract

Let C be a piecewise-smooth closed-convex curve in the plane R 2. A billiard ball is a point interior to C which moves with unit velocity until it hits a point p E C. If C has a tangent at p the ball bounces in the direction determined by reflection off of the unique tangent line at p, again with unit velocity. The billiard path is then the well-determined orbit traced by the point in the course of time, where we shall neglect orbits which encounter any corner point of C. A billiard path is said to be periodic if the orbit is a closed polygon with finitely many vertices on C. The polygon need not be simple; for example, nonsimple star figures occur as billiard orbits in circles. It is of interest to determine if a class of convex curves (or, more generally, a class of convex figures in Rd) can be characterized by a corresponding property of the billiard paths within the figure. Indeed there are a number of results of this type, both recent and classical. The billiard characterizations of the conic sections have long been known. For an example of a more recent result, a closed convex polygon P is regular if and only if P contains a periodic billiard path P' which is similar to P (see [1]). Two billiard characterizations of curves of constant width are known, the more recent an elegant result found in [3]. The other characterization motivates the work of this paper and is obtained by simply restating the well-known double normal property: a smooth closed convex curve C has constant width if and only if there is a periodic billiard 2-gon (i. e., double segment) at each point of C. This characterization of curves of constant width raises a rather intriguing question: