Does a billiard orbit determine its (polygonal) table?

TL;DR

This paper introduces a new way to classify polygonal billiards based on the order of dense orbits, exploring how these equivalence classes relate to the shape of the billiard table.

Contribution

It defines and investigates an order equivalence relation on polygonal billiards, linking orbit patterns to the geometric structure of the tables.

Findings

New equivalence relation on polygonal billiards introduced

Characterization of orbit sequences and their relation to table geometry

Insights into how orbit patterns determine table shape

Abstract

We introduce a new equivalence relation on the set of all polygonal billiards. We say that two billiards (or polygons) are order equivalent if each of the billiards has an orbit whose footpoints are dense in the boundary and the two sequences of footpoints of these orbits have the same combinatorial order. We study this equivalence relation with additional regularity conditions on the orbit.