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.

中文翻译：

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来自交叉比的离散曲率和扭转

受多边形莫比乌斯不变细分方案的启发，我们研究了离散曲线的曲率概念，其中交叉比在我们所有的关键定义中都起着重要作用。使用特定的莫比乌斯不变点插入规则，类似于经典的四点方案，我们沿着离散曲线构造圆。渐近分析表明，随着采样密度的增加，在采样曲线上定义的这些圆会收敛到平滑的曲率圆。我们使用交叉比表达空间曲线的离散扭转，这不是莫比乌斯不变概念，并显示其类似于曲率的渐近行为。