Loci of 3-periodics in an Elliptic Billiard: why so many ellipses?
Dynamical Systems
2022-10-11 v4 Computational Geometry
Robotics
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 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
Keywords
Cite
@article{arxiv.2001.08041,
title = {Loci of 3-periodics in an Elliptic Billiard: why so many ellipses?},
author = {Ronaldo Garcia and Jair Koiller and Dan Reznik},
journal= {arXiv preprint arXiv:2001.08041},
year = {2022}
}
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26 pages, 16 Figures, 6 Tables, 13 videos