A unified approach to Penner, Ptolemy, and Casey's theorems in several dimensions
Fuente:
arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2026
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| _version_ | 1866913155444113408 |
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| author | Lewis, Isabella Short, Ian |
| author_facet | Lewis, Isabella Short, Ian |
| contents | We prove Penner's theorem on horocycles and theorems of Ptolemy and Casey, all with full converses, in hyperbolic space of several dimensions. Recently Waddle observed that the equations underpinning these three theorems are related, and it is this viewpoint that we advance, using the Lorentzian model of hyperbolic space. We show that all three theorems can be derived from a common Gram-matrix calculation applied to lightlike, timelike, and spacelike vectors. Remarkably, our approach gives a version of Casey's theorem in the plane with a full converse, involving three geometric alternatives, which to our knowledge has not previously been recorded. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_23430 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | A unified approach to Penner, Ptolemy, and Casey's theorems in several dimensions Lewis, Isabella Short, Ian Metric Geometry Primary 51M10, 51M04, Secondary 30F60, 51B10 We prove Penner's theorem on horocycles and theorems of Ptolemy and Casey, all with full converses, in hyperbolic space of several dimensions. Recently Waddle observed that the equations underpinning these three theorems are related, and it is this viewpoint that we advance, using the Lorentzian model of hyperbolic space. We show that all three theorems can be derived from a common Gram-matrix calculation applied to lightlike, timelike, and spacelike vectors. Remarkably, our approach gives a version of Casey's theorem in the plane with a full converse, involving three geometric alternatives, which to our knowledge has not previously been recorded. |
| title | A unified approach to Penner, Ptolemy, and Casey's theorems in several dimensions |
| topic | Metric Geometry Primary 51M10, 51M04, Secondary 30F60, 51B10 |
| url | https://arxiv.org/abs/2605.23430 |