Code & order in polygonal billiards

TL;DR

This paper investigates the relationship between polygons that share the same billiard code sequences, revealing conditions under which they have identical angles, are similar, or affinely similar.

Contribution

It establishes new results linking code equivalence of polygons to their geometric similarity and affine properties.

Findings

Code equivalent polygons with dense boundary orbits have the same angles.

Such polygons are similar under certain conditions.

They are affinely similar in specific cases.

Abstract

Two polygons $P,Q$ are code equivalent if there are billiard orbits $u,v$ which hit the same sequence of sides and such that the projections of the orbits are dense in the boundaries $\partial P, \partial Q$. Our main results show when code equivalent polygons have the same angles, resp. are similar, resp. affinely similar.