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Analysis and design of discrete normals and curvatures


Langer,  Torsten
Computer Graphics, MPI for Informatics, Max Planck Society;

Belyaev,  Alexander
Computer Graphics, MPI for Informatics, Max Planck Society;

Seidel,  Hans-Peter
Computer Graphics, MPI for Informatics, Max Planck Society;

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Langer, T., Belyaev, A., & Seidel, H.-P.(2005). Analysis and design of discrete normals and curvatures (MPI-I-2005-4-003). Saarbrücken: Max-Planck-Institut für Informatik.

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Accurate estimations of geometric properties of a surface (a curve) from its discrete approximation are important for many computer graphics and computer vision applications. To assess and improve the quality of such an approximation we assume that the smooth surface (curve) is known in general form. Then we can represent the surface (curve) by a Taylor series expansion and compare its geometric properties with the corresponding discrete approximations. In turn we can either prove convergence of these approximations towards the true properties as the edge lengths tend to zero, or we can get hints how to eliminate the error. In this report we propose and study discrete schemes for estimating the curvature and torsion of a smooth 3D curve approximated by a polyline. Thereby we make some interesting findings about connections between (smooth) classical curves and certain estimation schemes for polylines. Furthermore, we consider several popular schemes for estimating the surface normal of a dense triangle mesh interpolating a smooth surface, and analyze their asymptotic properties. Special attention is paid to the mean curvature vector, that approximates both, normal direction and mean curvature. We evaluate a common discrete approximation and show how asymptotic analysis can be used to improve it. It turns out that the integral formulation of the mean curvature \begin{equation*} H = \frac{1}{2 \pi} \int_0^{2 \pi} \kappa(\phi) d\phi, \end{equation*} can be computed by an exact quadrature formula. The same is true for the integral formulations of Gaussian curvature and the Taubin tensor. The exact quadratures are then used to obtain reliable estimates of the curvature tensor of a smooth surface approximated by a dense triangle mesh. The proposed method is fast and often demonstrates a better performance than conventional curvature tensor estimation approaches. We also show that the curvature tensor approximated by our approach converges towards the true curvature tensor as the edge lengths tend to zero.