Loci of Triangular Orbits in an Elliptic Billiard

01/22/2020

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by   Dan Reznik, et al.

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We analyze the family of 3-periodic trajectories in an Elliptic Billiard. Taken as a continuum of rotating triangles, 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. Finally, we informally describe a variety of delightful phenomena involving loci of orbit-derived centers and vertices.