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arXiv · 2001.08041

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

Ronaldo Garcia·Jair Koiller·Dan Reznik
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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

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Ronaldo Garcia, Jair Koiller, Dan Reznik. 2020-01-22. Loci of 3-periodics in an Elliptic Billiard: why so many ellipses?. https://doi.org/10.1016/j.jsc.2022.06.001

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