Notes on Beltrami's Essay
Fuente:
arXiv
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| Formato: | Preprint |
| Publicado: |
2026
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| _version_ | 1866915994022182912 |
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| author | Rose, Steven |
| author_facet | Rose, Steven |
| contents | Eugenio Beltrami published his seminal 'Essay on the Interpretation of Non-Euclidean Geometry' in 1868, where he showed that geodesics on a surface of constant negative curvature can be mapped as straight lines on a Euclidean disc. More importantly he showed that figures on the disc would satisfy the identities of hyperbolic geometry characteristic of a surface of negative curvature. However Beltrami did not always give a full explanation of the equations which he used. These notes are an attempt to provide a derivation of some of his principal results, including his formula for hyperbolic distance on the disc, his proof that the sum of the (hyperbolic) angles of a triangle on the disc is less than two right angles and his equations for circles, equidistants and horocycles. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_07932 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Notes on Beltrami's Essay Rose, Steven History and Overview 01A55 Eugenio Beltrami published his seminal 'Essay on the Interpretation of Non-Euclidean Geometry' in 1868, where he showed that geodesics on a surface of constant negative curvature can be mapped as straight lines on a Euclidean disc. More importantly he showed that figures on the disc would satisfy the identities of hyperbolic geometry characteristic of a surface of negative curvature. However Beltrami did not always give a full explanation of the equations which he used. These notes are an attempt to provide a derivation of some of his principal results, including his formula for hyperbolic distance on the disc, his proof that the sum of the (hyperbolic) angles of a triangle on the disc is less than two right angles and his equations for circles, equidistants and horocycles. |
| title | Notes on Beltrami's Essay |
| topic | History and Overview 01A55 |
| url | https://arxiv.org/abs/2605.07932 |