Mannheim-d'Ocagne-Koenderink type formulas for asymptotic directions
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arXiv
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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866916538811940864 |
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| author | Fukui, Toshizumi Honda, Atsufumi Umehara, Masaaki |
| author_facet | Fukui, Toshizumi Honda, Atsufumi Umehara, Masaaki |
| contents | We consider a surface embedded in the Euclidean 3-space and fix a tangential vector $v$ at a given point $p$ on the surface. In this paper, we first review a history of the formula obtained by Mannheim, d'Ocagne and Koenderink, which asserts that the Gaussian curvature of the surface at $p$ can be obtained if one knows "the normal curvature at $p$ with respect to $v$" and "the curvature of the contour line $L$ of the surface at $p$" with respect to the orthogonal projection induced by $v$. Unfortunately, this formula does not work when $v$ points in an asymptotic direction. When $v$ is just the case, we give anlogues of the formula, which include an invariant of cusp singular points of $L$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_17244 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Mannheim-d'Ocagne-Koenderink type formulas for asymptotic directions Fukui, Toshizumi Honda, Atsufumi Umehara, Masaaki Differential Geometry We consider a surface embedded in the Euclidean 3-space and fix a tangential vector $v$ at a given point $p$ on the surface. In this paper, we first review a history of the formula obtained by Mannheim, d'Ocagne and Koenderink, which asserts that the Gaussian curvature of the surface at $p$ can be obtained if one knows "the normal curvature at $p$ with respect to $v$" and "the curvature of the contour line $L$ of the surface at $p$" with respect to the orthogonal projection induced by $v$. Unfortunately, this formula does not work when $v$ points in an asymptotic direction. When $v$ is just the case, we give anlogues of the formula, which include an invariant of cusp singular points of $L$. |
| title | Mannheim-d'Ocagne-Koenderink type formulas for asymptotic directions |
| topic | Differential Geometry |
| url | https://arxiv.org/abs/2412.17244 |