Periodic billiard trajectories in polyhedra

TL;DR

This paper studies periodic billiard trajectories within polyhedra, providing stability conditions, and demonstrates the existence of specific periodic orbits in tetrahedra, extending classical results to three dimensions.

Contribution

It introduces a stability criterion for periodic billiard trajectories in polyhedra and proves the existence of a family of tetrahedra with specific periodic orbits.

Findings

Established a stability condition for periodic trajectories.

Proved the existence of an open set of tetrahedra with a 4-periodic orbit.

Analyzed the orbit of points along the periodic coding.

Abstract

We consider the billiard map inside a polyhedron. We give a condition for the stability of the periodic trajectories. We apply this result to the case of the tetrahedron. We deduce the existence of an open set of tetrahedra which have a periodic orbit of length four (generalization of Fagnano's orbit for triangles), moreover we can study completly the orbit of points along this coding.