A discrete analog of Segre's theorem on spherical curves
Fuente:
arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2022
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| _version_ | 1866912194859368448 |
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| author | Gentil, Samuel Pacitti Craizer, Marcos |
| author_facet | Gentil, Samuel Pacitti Craizer, Marcos |
| contents | We prove a discrete analog of a certain four-vertex theorem for space curves. The smooth case goes back to the work of Beniamino Segre and states that a closed and smooth curve whose tangent indicatrix has no self-intersections admits at least four points at which its torsion vanishes. Our approach uses the notion of discrete tangent indicatrix of a (closed) polygon. Our theorem then states that a polygon with at least four vertices and whose discrete tangent indicatrix has no self-intersections admits at least four flattenings, i.e., triples of vertices such that the preceding and following vertices are on the same side of the plane spanned by this triple. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2208_04454 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | A discrete analog of Segre's theorem on spherical curves Gentil, Samuel Pacitti Craizer, Marcos Differential Geometry We prove a discrete analog of a certain four-vertex theorem for space curves. The smooth case goes back to the work of Beniamino Segre and states that a closed and smooth curve whose tangent indicatrix has no self-intersections admits at least four points at which its torsion vanishes. Our approach uses the notion of discrete tangent indicatrix of a (closed) polygon. Our theorem then states that a polygon with at least four vertices and whose discrete tangent indicatrix has no self-intersections admits at least four flattenings, i.e., triples of vertices such that the preceding and following vertices are on the same side of the plane spanned by this triple. |
| title | A discrete analog of Segre's theorem on spherical curves |
| topic | Differential Geometry |
| url | https://arxiv.org/abs/2208.04454 |