Loci of 3-periodics in an Elliptic Billiard: why so many ellipses?

Abstract

We analyze the family of 3-periodic (triangular) trajectories in an Elliptic Billiard. Specifically, the loci of their Triangle Centers such as the Incenter, Barycenter, etc. Many points have ellipses as loci, but some are also quartics, self-intersecting curves of higher degree, and even a stationary point. Elegant proofs have surfaced for locus ellipticity of a few classic centers, however these are based on laborious case-by-case analysis. Here we present two rigorous methods to detect when any given Center produces an elliptic locus: a first one which is a hybrid of numeric and computer algebra techniques (good for fast detection only), and a second one based on the Theory of Resultants, which computes the implicit two-variable polynomial whose zero set contains the locus.Comment: 32 pages, 14 Figures, 6 Tables, 13 video