On the angular defect of triangulations and the pointwise approximation of curvatures Original Research Article

Let S be a smooth surface of E3, p a point on S, km, kM, kG and kH 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 pn+1=p1. For each triangle, let γi be the angle ∠pippi+1, and let the angular defect at p be 2π−∑iγi. This paper establishes, when the distances ∥ppi∥ go to zero, that the angular defect is asymptotically equivalent to a homogeneous polynomial of degree two in the principal curvatures.