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Discrete curvature and torsion from cross-ratios
Christian Müller1, Amir Vaxman2
1Institute of Discrete Mathematics and Geometry, TU Wien, Wiedner Hauptstraße 8-10/104, 1040 Vienna, Austria.
Annali Di Matematica Pura Ed Applicata
|November 1, 2021
Summary
Researchers developed a new discrete curve curvature using the cross-ratio, inspired by Möbius transformations. This method constructs circles on discrete curves that approximate smooth curvature as sampling increases, also defining discrete torsion.
Area of Science:
- Differential Geometry
- Discrete Differential Geometry
- Geometric Analysis
Background:
- Möbius invariant subdivision schemes for polygons.
- Need for curvature notions in discrete settings.
- Role of cross-ratio in geometric definitions.
Purpose of the Study:
- Introduce a new curvature notion for discrete curves based on the cross-ratio.
- Develop a Möbius invariant point-insertion rule for discrete curves.
- Analyze the asymptotic behavior of discrete curvature and torsion.
Main Methods:
- Utilizing a Möbius invariant point-insertion rule analogous to the four-point scheme.
- Constructing discrete circles along curves.
- Performing asymptotic analysis on sampled curves.
- Expressing discrete torsion using the cross-ratio.
Main Results:
- Defined discrete curvature and circles using the cross-ratio.
- Demonstrated convergence of discrete curvature circles to smooth curvature circles with increased sampling density.
- Formulated discrete torsion for space curves via the cross-ratio.
- Showcased asymptotic behavior of discrete torsion analogous to curvature.
Conclusions:
- The cross-ratio provides a powerful tool for defining curvature and torsion in discrete settings.
- The proposed discrete curvature is a valid generalization of smooth curvature.
- The discrete torsion formulation offers new insights into the geometry of discrete space curves.
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