Magic Billiards: the Case of Elliptical Boundaries

TL;DR

This paper introduces the concept of magic billiards, unifying various billiard generalizations, and analyzes their properties with a focus on elliptical boundaries, providing conditions for periodicity and a topological classification.

Contribution

It presents a new unified framework for billiards called magic billiards and offers explicit conditions for periodicity in elliptical cases, along with a topological analysis.

Findings

Explicit conditions for periodicity in elliptical magic billiards

Topological classification using Fomenko graphs

Unification of several billiard generalizations

Abstract

In this work, we introduce a novel concept of magic billiards, which can be seen as an umbrella, unifying several well-known generalisations of mathematical billiards. We analyse properties of magic billiards in the case of elliptical boundaries. We provide explicit conditions for periodicity in algebro-geometric, analytic, and polynomial forms. A topological description of those billiards is given using Fomenko graphs.