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Concentrated Curvature for Mean Curvature Estimation
No full textWe present a mathematical result that allows computing the discrete mean curvature of a polygonal surface from the so-called concentrated curvature generally used for Gaussian curvature estimation. Our result adds important value to concentrated curvature as a geometric and metric tool to study accurately the morphology of a surface
Discrete distortion in triangulated 3-manifolds
No full textWe introduce a novel notion, that we call discrete distortion, for a triangulated 3-manifold. Discrete distortion naturally generalizes the notion of concentrated curvature defined for triangulated surfaces and provides a powerful tool to understand the local geometry and topology of 3-manifolds. Discrete distortion can be viewed as a discrete approach to Ricci curvature for singular flat manifolds. We distinguish between two kinds of distortion, namely, vertex distortion, which is associated with the vertices of the tetrahedral mesh decomposing the 3-manifold, and bond distortion, which is associated with the edges of the tetrahedral mesh. We investigate properties of vertex and bond distortions. As an example, we visualize vertex distortion on manifold hypersurfaces in R4 defined by a scalar field on a 3D mesh
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