Curvatures Estimation And The Improvement Of Taubin's Method On Triangular Mesh

Curvature is an important invariant of surface and the important foundation of differential geometry. At the same time,diserete curvature on triangular mesh is the foundation of many applications such as characteristic detecting on a polyhedral surface, smoothing, simplification, distortion and region decomposition.But the triangular mesh is defined by points cloud and topologic relation among the points,lack of definite analytic expression of surfaces, so the curvature formulas in differential geometry are not suitable for the discrete forms.So many diserete curvature estimation methods on triangular mesh appear.So far, there are about 10 diserete curvature estimation methods on triangular. The paper presents a systematic of various kinds of diserete curvature estimation methods on triangular mesh. Then wo introduce the theoretical background,the signification of formulas and the applied fields of several main methods in detail.Taubin's diserete curvature estimation methods plays an important role,which is linear in time and space. They described a method to estimate the tensor of curvature of a surface at the vertices of a triangular mesh. Principal curvatures and principal directions are obtained by computing in closed form the eigenvalues and eigenvectors of certain 3×3 symmetric matrices defined by integral formulas, and closely related to the matrix representation of the tensor of curvature.Taubin's diserete curvature estimation methods uses two intermediate quantities: weight and normal vectors.Taubin estimated normal vectors with area weights, which neglected the effects of triangular shape. Sheng-Gwo Chen presented the method of normal vectors and curvatures with centroid weights. We adopt area,centroid and angle weighted triangle normal vectors calculation formula and triangle centroid weights to improve Taubin's diserete curvature estimation methods, which more exactly reflect the effects of triangular shape.The paper provides particular comparison results of error on an parametric surface.It prove our method's advantage on the ellipsiod and torus. Compared with old Taubin's method, Gaussian and mean curvatures has its obvious advantage in exactitude.At last, we make an assay of errors. Our conclusions are curvature errors have much to do with points density and bend level of surface at the point.