Cusps of caustics by reflection in ellipses
Fuente:
arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866916289634631680 |
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| author | Bor, Gil Spivakovsky, Mark Tabachnikov, Serge |
| author_facet | Bor, Gil Spivakovsky, Mark Tabachnikov, Serge |
| contents | This paper is concerned with the billiard version of Jacobi's last geometric statement and its generalizations. Given a non-focal point $O$ inside an elliptic billiard table, one considers the family of rays emanating from $O$ and the caustic $Γ_n$ of the reflected family after $n$ reflections off the ellipse, for each positive integer $n$. It is known that $Γ_n$ has at least four cusps and it has been conjectured that it has exactly four (ordinary) cusps. The present paper presents a proof of this conjecture in the special case when the ellipse is a circle. In the case of an arbitrary ellipse, we give an explicit description of the location of four of the cusps of $Γ_n$, though we do not prove that these are the only cusps. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_11074 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Cusps of caustics by reflection in ellipses Bor, Gil Spivakovsky, Mark Tabachnikov, Serge Differential Geometry 53A04 This paper is concerned with the billiard version of Jacobi's last geometric statement and its generalizations. Given a non-focal point $O$ inside an elliptic billiard table, one considers the family of rays emanating from $O$ and the caustic $Γ_n$ of the reflected family after $n$ reflections off the ellipse, for each positive integer $n$. It is known that $Γ_n$ has at least four cusps and it has been conjectured that it has exactly four (ordinary) cusps. The present paper presents a proof of this conjecture in the special case when the ellipse is a circle. In the case of an arbitrary ellipse, we give an explicit description of the location of four of the cusps of $Γ_n$, though we do not prove that these are the only cusps. |
| title | Cusps of caustics by reflection in ellipses |
| topic | Differential Geometry 53A04 |
| url | https://arxiv.org/abs/2406.11074 |