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

Let S be a smooth surface of E³, 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 = p₁. For each triangle, let γ_i be the angle ∠p_ipp _i+1, and let the angular defect at p be 2π - ∑ _iγ_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. 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.