Billiard algebra, integrable line congruences, and double reflection nets

TL;DR

This paper explores the integrability of billiard systems within quadrics, introducing double-reflection nets and their relation to line congruences, Yang-Baxter maps, and integrable quad-graphs, advancing the mathematical understanding of discrete billiard dynamics.

Contribution

It introduces the concept of double-reflection nets linked to pencils of quadrics and establishes their connection to integrability conditions and Yang-Baxter maps.

Findings

The Six-pointed star theorem is equivalent to an integrability condition.

Double-reflection nets are a subclass of dual Darboux nets associated with quadrics.

Yang-Baxter maps related to pencils of quadrics are defined and analyzed.

Abstract

The billiard systems within quadrics, playing the role of discrete analogues of geodesics on ellipsoids, are incorporated into the theory of integrable quad-graphs. An initial observation is that the Six-pointed star theorem, as the operational consistency for the billiard algebra, is equivalent to an integrabilty condition of a line congruence. A new notion of the double-reflection nets as a subclass of dual Darboux nets associated with pencils of quadrics is introduced, basic properies and several examples are presented. Corresponding Yang-Baxter maps, associated with pencils of quadrics are defined and discussed.