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Discrete Curvature and Torsion from Cross-Ratios
Motivated by a M\"obius 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 M\"obius 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 M\"obius invariant notion, using the cross-ratio and
show its asymptotic behavior in analogy to the curvature