Discrete curve theory in space forms: planar elastic and area-constrained elastic curves
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arXiv
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| Auteurs principaux: | , , |
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866909464334958592 |
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| author | Hoffmann, Tim Steinmeier, Jannik Szewieczek, Gudrun |
| author_facet | Hoffmann, Tim Steinmeier, Jannik Szewieczek, Gudrun |
| contents | We propose a notion of discrete elastic and area-constrained elastic curves in 2-dimensional space forms. Our definition extends the well-known discrete Euclidean curvature equation to space forms and reflects various geometric properties known from their smooth counterparts. Special emphasis is paid to discrete flows built from Bäcklund transformations in the respective space forms. The invariants of the flows form a hierarchy of curves and we show that discrete elastic and constrained elastic curves can be characterized as elements of this hierarchy. This work also includes an introductory chapter on discrete curve theory in space forms, where we find discrete Frenet-type formulas and describe an associated family related to a fundamental theorem. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_13477 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Discrete curve theory in space forms: planar elastic and area-constrained elastic curves Hoffmann, Tim Steinmeier, Jannik Szewieczek, Gudrun Differential Geometry 53A70 (primary) 53A35 53E40 (secondary) We propose a notion of discrete elastic and area-constrained elastic curves in 2-dimensional space forms. Our definition extends the well-known discrete Euclidean curvature equation to space forms and reflects various geometric properties known from their smooth counterparts. Special emphasis is paid to discrete flows built from Bäcklund transformations in the respective space forms. The invariants of the flows form a hierarchy of curves and we show that discrete elastic and constrained elastic curves can be characterized as elements of this hierarchy. This work also includes an introductory chapter on discrete curve theory in space forms, where we find discrete Frenet-type formulas and describe an associated family related to a fundamental theorem. |
| title | Discrete curve theory in space forms: planar elastic and area-constrained elastic curves |
| topic | Differential Geometry 53A70 (primary) 53A35 53E40 (secondary) |
| url | https://arxiv.org/abs/2501.13477 |