On surface normal and Gaussian curvature approximations given data sampled from a smooth surface
Computer Aided Geometric Design
Smooth surface and triangular mesh: comparison of the area, the normals and the unfolding
Proceedings of the seventh ACM symposium on Solid modeling and applications
Optimizing 3D triangulations using discrete curvature analysis
Mathematical Methods for Curves and Surfaces
Computation of Local Differential Parameters on Irregular Meshes
Proceedings of the 9th IMA Conference on the Mathematics of Surfaces
Estimating the tensor of curvature of a surface from a polyhedral approximation
ICCV '95 Proceedings of the Fifth International Conference on Computer Vision
Convergence analysis of a discretization scheme for Gaussian curvature over triangular surfaces
Computer Aided Geometric Design
ACM Transactions on Mathematical Software (TOMS)
Discrete schemes for Gaussian curvature and their convergence
Computers & Mathematics with Applications
Computer Aided Geometric Design
Convergence analysis of a discretization scheme for Gaussian curvature over triangular surfaces
Computer Aided Geometric Design
Illustrative visualization: interrogating triangulated surfaces
Computing - Geometric Modelling, Dagstuhl 2008
Curvature estimation over smooth polygonal meshes using the half tube formula
Proceedings of the 12th IMA international conference on Mathematics of surfaces XII
Error term in pointwise approximation of the curvature of a curve
Computer Aided Geometric Design
Generalized shape operators on polyhedral surfaces
Computer Aided Geometric Design
Polyhedral gauss maps and curvature characterisation of triangle meshes
IMA'05 Proceedings of the 11th IMA international conference on Mathematics of Surfaces
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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 - Σiγi. 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.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 kG 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 kG within an error bound depending upon km and kM.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.