Abstract
To a closed convex smooth curve in the plane the dual billiard transformation of its exterior corresponds: given a point outside of the curve, draw a tangent line to it through the point, and reflect the point in the point of tangency. We prove that if two curves are given, such that the corresponding dual billiard transformations commute, then the curves are concentric homothetic ellipses.
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Tabachnikov, S. Commuting dual billiard maps. Geom Dedicata 53, 57–68 (1994). https://doi.org/10.1007/BF01264044
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DOI: https://doi.org/10.1007/BF01264044