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

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 {pippi+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 anglepipp(i+1), and let the angular defect at p be 2pi - Sigma(i) gammai. This paper establishes, when the distances parallel toppiparallel to 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). Meshes are ubiquitous in Computer Graphics and Computer Aided Design, and a significant number of papers advocate the use of normalized angular defects to estimate the Gauss curvature of smooth surfaces. We show that the statements made in these papers are erroneous in general, although they may be true pointwise for very specific meshes. A direct consequence is that normalized angular defects should be used to estimate the Gauss curvature for these cases only where the geometry of the meshes processed is precisely controlled. On a more general perspective, we believe this contributions is one step forward the intelligence of the geometry of meshes, whence one step forward more robust algorithms. (C) 2003 Elsevier B.V. All rights reserved.