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.
Highlights
Many topics in applied geometry like computer graphics, computer vision, and geometry processing in general, cover tasks like the acquisition and analysis of geometric data, its reconstruction, and further its manipulation and simulation
A different approach to discrete curvatures comes from discrete differential geometry [2]; the motivation behind any discretization is to apply the ideas and methods from classical differential geometry instead of “” discretizing equations or using classical differential
Discrete curvatures defined via this approach are connected to a sensible notion of a curvature circle [9], a consistent definition of a Frenet-frame [5], or geometric ideas that appear in geometric knot theory [14]
Summary
Many topics in applied geometry like computer graphics, computer vision, and geometry processing in general, cover tasks like the acquisition and analysis of geometric data, its reconstruction, and further its manipulation and simulation. Discrete curvatures defined via this approach are connected to a sensible notion of a curvature circle [9], a consistent definition of a Frenet-frame [5], or geometric ideas that appear in geometric knot theory [14]. In analogy to Sauer [13], we discretize/sample a smooth curve s(t) by constructing the inscribed polygon s(k ) with k ∈ Z as depicted in Fig. 2 (right) Using this discrete curve, we prove that our discrete curvature k , which is defined at the polygon edge k, k + 1 , is a second-order approximation of the curvature of s, i.e., k = + O( 2) as → 0 (see Theorem 2). From our definition of the curvature circle we immediately obtain a discrete Frenet-frame in Theorem 3 and Sect. The Möbius invariance of the cross-ratio implies the same for the curvature circle, analogous to smooth curves.
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