Loci of 3periodics in an Elliptic Billiard: why so many ellipses?
Abstract
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 3periodics in the elliptic billiard, and the loci of its triangle centers. Some will sweep ellipses, and others higherdegree 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 algebrogeometric method. We also prove that if the triangle center function is rational on sidelengths, the locus is algebraic
 Publication:

arXiv eprints
 Pub Date:
 January 2020
 arXiv:
 arXiv:2001.08041
 Bibcode:
 2020arXiv200108041G
 Keywords:

 Mathematics  Dynamical Systems;
 Computer Science  Computational Geometry;
 Computer Science  Robotics;
 3740;
 51N20;
 51M04;
 5104
 EPrint:
 26 pages, 16 Figures, 6 Tables, 13 videos