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arXiv · 2501.04867

On cusps of caustics by reflection in two dimensional projective Finsler metrics

Serge Tabachnikov
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Abstract

A Finsler, not necessarily symmetric, metric in the plane or its convex subset is called projective if its geodesics are straight segments. We consider Finsler billiards in a convex planar domain endowed with a projective Finsler metric. A caustic by reflection is the envelope of the oriented lines, the billiard trajectories, that start at a point inside the billiard and undergo a fixed number of reflections. We show that such a caustic has at least four cusps. This problem is motivated by the "Last Geometric Statement of Jacobi" that the conjugate locus of a non-umbilic point of a triaxial ellipsoid has exactly four cusps. The present note extends the recent results in this direction concerning Euclidean billiards.

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Serge Tabachnikov. 2025-01-08. On cusps of caustics by reflection in two dimensional projective Finsler metrics. https://arxiv.org/abs/2501.04867

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