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Complex Caustics of the Elliptic Billiard

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Abstract

The article studies a generalization of the elliptic billiard to the complex domain. We show that the billiard orbits also have caustics, and that the number of such caustics is bigger than for the real case. For example, for a given ellipse E, there exist exactly two confocal ellipses such that the triangular orbits of E are circumscribed about one of them, and each tangent line to one of those ellipses is a side of a triangular orbit. In the case of 4-periodic orbits, we get generically three caustics. We also give an upper bound on the number of caustics for orbits with a fixed number of sides, and explain how to compute its exact value.

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  1. École Normale Supérieure de Lyon, Unité de Mathématiques Pures et Appliquées, UMR CNRS 5669, 46, allée d’Italie, 69364, Lyon Cedex 07, France

    Corentin Fierobe

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Fierobe, C. Complex Caustics of the Elliptic Billiard. Arnold Math J. 7, 1–30 (2021). https://doi.org/10.1007/s40598-020-00152-w

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  • DOI: https://doi.org/10.1007/s40598-020-00152-w

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