Discrete Curvature and Torsion from Cross-Ratios
Abstract
Motivated by a Möbius invariant subdivision scheme for polygons, we study a curvature notion for discrete curves where the cross-ratio plays an important role in all our key definitions. Using a particular Möbius invariant point-insertion-rule, comparable to the classical four-point-scheme, we construct circles along discrete curves. Asymptotic analysis shows that these circles defined on a sampled curve converge to the smooth curvature circles as the sampling density increases. We express our discrete torsion for space curves, which is not a Möbius invariant notion, using the cross-ratio and show its asymptotic behavior in analogy to the curvature.
- Publication:
-
arXiv e-prints
- Pub Date:
- August 2020
- DOI:
- 10.48550/arXiv.2008.13236
- arXiv:
- arXiv:2008.13236
- Bibcode:
- 2020arXiv200813236M
- Keywords:
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- Mathematics - Differential Geometry;
- Computer Science - Graphics;
- Mathematics - Numerical Analysis