Loci of 3-periodics 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 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

26 pages, 16 Figures, 6 Tables, 13 videos

References in corpus (2)

• Eighty New Invariants of N-Periodics in the Elliptic Billiard
• The Ballet of Triangle Centers on the Elliptic Billiard

Cited by in corpus (9)

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• The Ballet of Triangle Centers on the Elliptic Billiard
• Parable of the Parabola
• Loci of the Brocard Points over Selected Triangle Families
• The Talented Mr. Inversive Triangle in the Elliptic Billiard
• Elliptic billiard with harmonic potential: Classical description
• Intriguing Invariants of Centers of Ellipse-Inscribed Triangles
• The Circumbilliard: Any Triangle can be a 3-Periodic
• Circum- and Inconic Invariants of 3-Periodics in the Elliptic Billiard