Loci of 3-periodics in an Elliptic Billiard: why so many ellipses?
This work addresses a specific geometric problem in dynamical systems, providing incremental insights into the behavior of triangle centers in elliptic billiards.
The paper tackles the problem of determining when triangle centers of 3-periodic orbits in an elliptic billiard trace out ellipses, proposing two rigorous methods—one using computer algebra and another algebro-geometric—to prove such loci are ellipses, and establishes that rational center functions yield algebraic loci.
A triangle center such as the incenter, barycenter, etc., is specified by a function thrice- and cyclically applied on sidelengths and/or angles. Consider the 1d family of 3-periodics in the elliptic billiard, and the loci of its triangle centers. Some will sweep ellipses, and others higher-degree algebraic curves. We propose two rigorous methods to prove if the locus of a given center is an ellipse: one based on computer algebra, and another based on an algebro-geometric method. We also prove that if the triangle center function is rational on sidelengths, the locus is algebraic