Fagnano Orbits of Polygonal Dual Billiards

Abstract

Given a convex n-gon P , a Fagnano periodic orbit of the respective dual billiard map is an n-gon Q whose sides are bisected by the vertices of P. For which polygons P does the ratio AreaQ=AreaP have the minimal value? The answer is shown to be: for affine-regular polygons. Mathematics Subject Classification ( 1991): 52-XX. The aim of this note is to provide a dual billiard counterpart to E. Gutkin's results ((2)) on Fagnano billiard orbits. Although the results and proofs in this note are quite elementary, I hope to attract the reader's attention to the lesser known dy- namical system, the dual (a.k.a. outer) billiard which is, in many ways, similar to, and sometimes easier to study than the conventional billiard. One plays the game of dual billiard outside (rather than inside) the billiard table. Let the table be a convex polygon P in the plane and x a point in its exterior (but not on the continuation of either of its sides). There are two support lines to P through x; choose the right one, as viewed from x, and reflect x in the support vertex of P to obtain the image of x under the dual billiard map, denoted by T in Figure 1. Clearly, the dual billiard map is a central symmetry in each component of its domain. The infinite orbit of a point x under this map is defined if none of the images or preimages ofx belongs to the continuation of a side of the polygon P ; the set of such 'good' points has full measure. Notice that the dual billiard map is equivariant under affine transformations of the plane. The definition extends to other convex billiard tables (say, ovals); remarkably, the dual billiard map is area preserving. The interested reader is referred to the sur- veys (7-8) where, in particular, he will find an explanation of the duality between usual and dual billiards and a discussion of multi-dimensional dual billiards. Before going into discussion of Fagnano dual billiard orbits, the actual topic of the note, mention a few facts about polygonal dual billiards. If P is ar ational polygon then every orbit is periodic. For a wider class of polygons, described in (3, 6) and including affine-regular polygons, every orbit of the dual billiard map stays bounded, but it is not known whether this holds true for all polygons (see (5)). It is not known either whether the dual billiard map has periodic orbits for every polygon P. The orbits of the polygonal dual billiard map can be infinite as