Can the Elliptic Billiard Still Surprise Us?

TL;DR

This paper explores the rich geometric properties of the elliptic billiard, revealing new invariants and surprising behaviors in periodic trajectories, including generalizations to multiple edges and conserved ratios.

Contribution

It introduces novel invariants and geometric relations for periodic billiard trajectories within an elliptic table, extending known results to arbitrary numbers of edges.

Findings

Discovery of new geometric loci for 3-periodic trajectories.

Identification of conserved ratios such as Inradius-to-Circumradius.

Generalization of 3-periodic invariants to trajectories with more edges.

Abstract

Can any secrets still be shed by that much studied, uniquely integrable, Elliptic Billiard? Starting by examining the family of 3-periodic trajectories and the loci of their Triangular Centers, one obtains a beautiful and variegated gallery of curves: ellipses, quartics, sextics, circles, and even a stationary point. Secondly, one notices this family conserves an intriguing ratio: Inradius-to-Circumradius. In turn this implies three conservation corollaries: (i) the sum of bounce angle cosines, (ii) the product of excentral cosines, and (iii) the ratio of excentral-to-orbit areas. Monge's Orthoptic Circle's close relation to 4-periodic Billiard trajectories is well-known. Its geometry provided clues with which to generalize 3-periodic invariants to trajectories of an arbitrary number of edges. This was quite unexpected. Indeed, the Elliptic Billiard did surprise us!