Polygonal billiards and "optical tori"

TL;DR

This paper explores polygonal billiards through an optical physics perspective, linking billiard trajectories to complex analysis and establishing a correspondence with punctured tori to analyze periodic paths.

Contribution

It introduces a novel approach connecting polygonal billiards with complex analysis and torus geometry, providing new insights into periodic trajectories.

Findings

Establishes a correspondence between n-gon billiards and 2n-punctured tori

Relates periodic billiard trajectories to closed geodesics on associated tori

Uses optical physics and complex analysis to analyze billiard dynamics

Abstract

We give an optical physicist view of the problem of the trajectories in a polygonal billiard using only basic facts of Optics and the theory of functions of a complex variable. This approach allow us to stablish a certain correspondence between n-gon billiards and one-holed 2n-punctured tori. Therefore the existence of periodic trajectories in a certain polygon becomes the problem of the existence of closed geodesics in its associated torus.