# Reference for standard lemmas in polygonal billiards

I'm trying to track down references for a couple of well-known results in the theory of polygonal billiards for a paper I'm working on.

Recall that periodic trajectories on a polygonal table occur in parallel families (I've seen these called cylinders or pencils), and that to any periodic table we can associate the sequence of edges which the trajectory crosses (which I've seen called the combinatorial or orbit type). It is well-known that

Up to reparametrization, two periodic trajectories lie in the same cylinder iff they have the same combinatorial type.

But I've never seen a proof of this fact, or even an explicit statement of it in literature.

Similarly, to each combinatorial type of a periodic trajectory on a triangle we can associate the set of triangles which admit periodic trajectories of that combinatorial type, called an orbit tile. Viewing the set of triangles as a subset of the plane (with the two coordinates interpreted as two angles of the triangle), it is well-known that

An orbit tile is either open, an open subset of a line segment, or a point.

This is stated for example in Section 2.1 of "Obtuse Triangular Billiards II: 100 Degrees Worth of Periodic Trajectories" by R. E. Schwartz, but I've never seen a proof of this in literature, and was unable to track one down in Schwartz's references.

Does anyone know where I can find proofs of these facts, or even a statement of the first?

• How do you define two trajectories belong to the same family? – David Mahone Jun 7 '13 at 0:24
• @David If they are homotopic via a homotopy which does not have any vertex of the polygon in its image. – Alex Becker Jun 7 '13 at 0:55