Conversion of matrix weighted rational B\'{e}zier curves to rational B\'{e}zier curves

Matrix weighted rational B\'{e}zier curves can represent complex curve shapes using small numbers of control points and clear geometric definitions of matrix weights. Explicit formulae are derived to convert matrix weighted rational B\'{e}zier curves in 2D or 3D space to rational B\'{e}zier curves. A method for computing the convex hulls of matrix weighted rational B\'{e}zier curves is given as a conjecture.Comment: 6 pages, 2 figure

Anisotropic Mesh Filtering by Homogeneous MLS Fitting

In this paper we present a novel geometric filter, a homogeneous moving least squares fitting-based filter (H-MLS filter), for anisotropic mesh filtering. Instead of fitting the noisy data by a moving parametric surface and projecting the noisy data onto the surface, we compute new positions of mesh vertices as the solutions to homogeneous least squares fitting of moving constants to local neighboring vertices and tangent planes that pass through the vertices. The normals for defining the tangent planes need not be filtered beforehand but the parameters for balancing the influences between neighboring vertices and neighboring tangent planes are computed robustly from the original data under the assumption of quadratic precision in each tangent direction. The weights for respective neighboring points for the least squares fitting are computed adaptively for anisotropic filtering. The filter is easy to implement and has distinctive features for mesh filtering. (1) The filter is locally implemented and has circular precision, spheres and cylinders can be recovered exactly by the filter. (2) The filtered mesh has a high fidelity to the original data without any position constraint and salient or sharp features can be preserved well. (3) The filter can be used to filter meshes with various kinds of noise as well as meshes with highly irregular triangulation.Comment: 13pages, 9 figure