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On Wed, 28 Feb 2007 12:08:04 -0500, Bruce Fischl <[log in to unmask]> wrote: >for Gaussian curvature using the discrete methods is the way to go (it's >essentially how much less than 180 deg the angles of the triangle add up >to). I don't think you can get principal directions or curvatures this way. Hi Bruce, I've had a chat to Philip, and it sounds like it is possible to get the complete curvature tensor with discrete methods; he provided the following information: "[...] refers to some subtle issues: From a Discrete Geometry point of view: the Gauss Curvature is naturally defined at VERTICES: it's the angle defect (proportional to 2 pi - sum of angles there) but, the Mean curvature is naturally defined on EDGES: (it's proportional to the dihedral angle = angle between normals of facets containing that edge). Thus, in my code, I 'interpolated' the mean curvature from edges to vertices, i.e. attribute some mean curvature of the edge to its vertices. (check vtkCurvatures.cxx for details). The Gauss curvature is analytically: K = k_1 * k_2, the mean H = (k_1 + k_2)/2 thus you can solve for k_1, and k_2 (the principal curvatures), but it's not always good. At first I didn't know of another geometric (angle based) way to define the principal curvatures but I found a report by Taubin, who computed discretely the 'curvature tensor' at every vertex: from it you can compute eigenvectors/eigenvalues, which are principal directions/curvatures. [...]" I've found the Taubin reference here: http://citeseer.ist.psu.edu/taubin95estimating.html And a more recent related paper (which PB also recommends): http://citeseer.ist.psu.edu/meyer02discrete.html Philip's website has some more details, references, and code: http://www.cs.ucl.ac.uk/staff/p.batchelor/curvatures/curvatures.html Best, Ged