On Surfaces in R^n via Gauss Map, Caustics, Duality and Pseudo Euclidean Geometry of Quadratic Forms
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866915482339115008 |
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| author | Uribe-Vargas, Ricardo |
| author_facet | Uribe-Vargas, Ricardo |
| contents | We get new results (and rederive some know ones) on smooth surfaces in $\mathbb{R}^n$ by unifying several view points into a coherent general view. Namely, we show and use new relations of the evolute (caustic) with the curvature ellipse, the Gauss map and the pseudo-Euclidean geometry of the $3$-space of quadratic forms on $\mathbb{R}^2$. A key result (Th.3.3.1): for a surface $M$ in $\mathbb{R}^n$ the intersection of its caustic with the normal space $N_pM$ is the polar dual hypersurface (in $N_pM$) of the curvature ellipse at $p$. Moreover, all local objects $X$ (cf. the invariants and their relations) have a "paired" version $X^*$ (with ${X^*}^*=X$) -- this provides new results on the original objects. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_14104 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On Surfaces in R^n via Gauss Map, Caustics, Duality and Pseudo Euclidean Geometry of Quadratic Forms Uribe-Vargas, Ricardo Differential Geometry We get new results (and rederive some know ones) on smooth surfaces in $\mathbb{R}^n$ by unifying several view points into a coherent general view. Namely, we show and use new relations of the evolute (caustic) with the curvature ellipse, the Gauss map and the pseudo-Euclidean geometry of the $3$-space of quadratic forms on $\mathbb{R}^2$. A key result (Th.3.3.1): for a surface $M$ in $\mathbb{R}^n$ the intersection of its caustic with the normal space $N_pM$ is the polar dual hypersurface (in $N_pM$) of the curvature ellipse at $p$. Moreover, all local objects $X$ (cf. the invariants and their relations) have a "paired" version $X^*$ (with ${X^*}^*=X$) -- this provides new results on the original objects. |
| title | On Surfaces in R^n via Gauss Map, Caustics, Duality and Pseudo Euclidean Geometry of Quadratic Forms |
| topic | Differential Geometry |
| url | https://arxiv.org/abs/2504.14104 |