Normal Vectors And Curvatures Estimation On Triangular Mesh

Curvature is an important geometric attribute on surface. The implicit or parametric surface has been defined very well in differential geometry. The triangular mesh is defined by points cloud and topologic relation among the points. These formulas of the first and second order differential properties in differential geometry can not be used on a triangular mesh, and many estimation methods appear. At the same time, curvatures estimation is the foundation of many applications at arbitrary vertices on a triangular mesh such as characteristic detecting on a polyhedral surface, smoothing, simplification, distortion and region decomposition.Taubin presented a simple method to estimate curvatures, 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.However, we found out that Taubin estimated normal with area weights, which neglected the effects of triangular shape. Sheng-Gwo Chen presented the method of normal vectors and curvatures with centroid weights. Taubin used circular arcs to approximate the normal curvature, and Chen-Shi Dong described a more accurate method, which estimated the normal curvature with normal information of neighborhood points. In this paper, we use centroid weights to approximate the normal vectors and new normal curvatures estimation method to improve Taubin's method in 1995.We provide particular comparison results of error on an implicit and parametric surface. Normal vectors, Gaussian and mean curvatures estimation are obtained by different curvatures estimation methods, and the error is computed by comparing estimation results with real ones on the sphere of unit radius and toms. Compared with old Taubin's method, the improved Taubin's method has its advantage in stabilization and exactitude. We also compare the results of our method with Meyer's, and get better results under certain conditions for mean curvatures estimation.