We analyze the family of 3-periodic (triangular) trajectories in an Elliptic Billiard. Its Triangle Centers ( Incenter, Barycenter, etc. ) sweep remarkable loci: ellipses , circles, quartics, sextics, and even a stationary point.  Here we present a systematic method to prove 29 out of the first 100 Centers listed in Clark Kimberling's Encyclopedia are elliptic. We also derive conditions under which loci are algebraic .
We analyze the family of 3-periodic (triangular) trajectories in an Elliptic Billiard. Specifically, the loci of their Triangle Centers such as the Incenter, Barycenter, etc. Many points have ellipses as loci, but some are also quartics, self-intersecting curves of higher degree , and even a stationary point. Elegant proofs have surfaced for locus ellipticity of a few classic centers, however these are based on laborious case-by-case analysis. Here we present two rigorous methods to detect when any given Center produces an elliptic locus: a first one which is a hybrid of numeric and computer algebra techniques (good for fast detection only), and a second one based on the Theory of Resultants, which computes the implicit two-variable polynomial whose zero set contains the locus .
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Its Triangle Centers (
Specifically, the loci of their Triangle Centers such as the
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) sweep remarkable loci: ellipses
Many points have ellipses as loci, but some are also quartics, self-intersecting curves of higher degree
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circles, quartics, sextics,
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Elegant proofs have surfaced for locus ellipticity of a few classic centers, however these are based on laborious case-by-case analysis.
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a systematic method to prove 29 out of the first 100 Centers listed in Clark Kimberling's Encyclopedia are elliptic. We also derive conditions under which loci are algebraic
two rigorous methods to detect when any given Center produces an elliptic locus: a first one which is a hybrid of numeric and computer algebra techniques (good for fast detection only), and a second one based on the Theory of Resultants, which computes the implicit two-variable polynomial whose zero set contains the locus