Discrete Curvature and Torsion from Cross-Ratios

TL;DR

This paper introduces a M"obius invariant approach to defining discrete curvature and torsion for polygons using cross-ratios, with convergence properties to smooth curvature in dense sampling.

Contribution

It develops a novel M"obius invariant discrete curvature and torsion framework based on cross-ratios, including a circle construction method and asymptotic analysis.

Findings

Discrete circles converge to smooth curvature circles with increased sampling density

The proposed torsion measure behaves asymptotically similar to curvature

The method provides a M"obius invariant way to analyze discrete space curves

Abstract

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.