On the angular defect of triangulations and the pointwise approximation of curvatures

Vincent Borrelli 1 Frédéric Cazals Jean-Marie Morvan
1 PRISME - Geometry, Algorithms and Robotics
CRISAM - Inria Sophia Antipolis - Méditerranée
Abstract : Let S be a smooth surface of E^3, p a point on S, k_m, k_M, k_G and k_H the maximum, minimum, Gauss and mean curvatures of S at p. Consider a set {p_ipp_(i+1)} i=1,...,n of n Euclidean triangles forming a piecewise linear approximation of S around p ---with p_(n+1)=p_1. For each triangle, let [gamma]_i be the angle /_ p_ipp_(i+1), and let the angular defect at p be 2[pi]-[SIGMA]_i[gamma- ]_i. This paper establishes, when the distances || pp_i || go to zero, that the angular defect is asymptotically equivalent to a homogeneous polynomial of degree two in the principal curvatures. For regular meshes, we provide closed forms expressions for the three coefficients of this polynomial. We show that vertices of valence four and six are the only ones where k_G can be inferred from the angular defect. At other vertices, we show that the principal curvatures can be derived from the angular defects of two independent triangulations. For irregular meshes, we show that the angular defect weighted by the so-called module of the mesh estimates k_G within an error bound depending upon k_m and k_M. Since meshes are so ubiquitous in Computer Graphics and Computer Aided Design, and Discrete Differential operators are so necessary to their processing, we believe these contributions are one step forward the intelligence of the geometry of meshes, whence one step forward more robust algorithms.
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Submitted on : Tuesday, May 23, 2006 - 7:29:47 PM
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Vincent Borrelli, Frédéric Cazals, Jean-Marie Morvan. On the angular defect of triangulations and the pointwise approximation of curvatures. RR-4590, INRIA. 2002. ⟨inria-00071995⟩

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