Loci of 3-periodics in an Elliptic Billiard: Why so many ellipses?. (January 2023)

• Record Type: Journal Article
• Authors: Garcia, Ronaldo; Koiller, Jair; Reznik, Dan
• Is Part Of: Journal of symbolic computation. Volume 114(2023)
• Page Start: 336
• Page End: 358
• Publication Date: 2023-01
• Subjects: 37-40 -- 51N20 -- 51M04 -- 51-04; Poncelet -- Porism -- Locus -- Resultants -- Algebraic; Mathematics -- Data processing -- Periodicals; Numerical analysis -- Data processing -- Periodicals; Automatic programming (Computer science) -- Periodicals; Mathématiques -- Informatique -- Périodiques; Analyse numérique -- Informatique -- Périodiques; Programmation automatique -- Périodiques; Automatic programming (Computer science); Mathematics -- Data processing; Numerical analysis -- Data processing; Periodicals; Electronic journals; 510.285
• DOI: 10.1016/j.jsc.2022.06.001
• Languages: English
• ISSNs: 0747-7171

Abstract

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.