On cusps of caustics by reflection in two dimensional projective Finsler metrics
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866909452215517184 |
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| author | Tabachnikov, Serge |
| author_facet | Tabachnikov, Serge |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2501_04867 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On cusps of caustics by reflection in two dimensional projective Finsler metrics Tabachnikov, Serge Differential Geometry Dynamical Systems 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. |
| title | On cusps of caustics by reflection in two dimensional projective Finsler metrics |
| topic | Differential Geometry Dynamical Systems |
| url | https://arxiv.org/abs/2501.04867 |