A Correspondence between Billiards and Geodesics
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arXiv
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| Format: | Preprint |
| Published: |
2026
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| _version_ | 1866916071257145344 |
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| author | Giannetto, Daniele |
| author_facet | Giannetto, Daniele |
| contents | From a geometric viewpoint, billiard trajectories and geodesics are related by mutual approximation results. In one direction, it is known that every geodesic curve on the boundary of a smooth convex body can be approximated by a sequence of billiard trajectories inside of it. We establish the other direction by proving that, for Riemannian billiard tables (under mild assumptions), there are families of fold-type surfaces such that every sequence of geodesic segments on these surfaces has a subsequence that converges to a billiard trajectory in the table. In particular, this is true for convex Euclidean tables. We also describe a more general class of tables for which this result holds and present explicit non-convex examples. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_02938 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | A Correspondence between Billiards and Geodesics Giannetto, Daniele Differential Geometry From a geometric viewpoint, billiard trajectories and geodesics are related by mutual approximation results. In one direction, it is known that every geodesic curve on the boundary of a smooth convex body can be approximated by a sequence of billiard trajectories inside of it. We establish the other direction by proving that, for Riemannian billiard tables (under mild assumptions), there are families of fold-type surfaces such that every sequence of geodesic segments on these surfaces has a subsequence that converges to a billiard trajectory in the table. In particular, this is true for convex Euclidean tables. We also describe a more general class of tables for which this result holds and present explicit non-convex examples. |
| title | A Correspondence between Billiards and Geodesics |
| topic | Differential Geometry |
| url | https://arxiv.org/abs/2602.02938 |