A Symmetric And Non-uniform Subdivision Algorithm Near Extraordinary Points

Non-Uniform Rational B-Splines (NURBS) and subdivision surfaces aretwo main technologies to design and represent freeform surfaces, both of themare based on uniform B-splines. NURBS are the dominant standard for CAD,while subdivision surfaces are only popularly used in animation andentertainment. However, subdivision surfaces could deal with arbitrarytopology mesh, which would be useful within Computer-Aided Design.Meanwhile, arbitrary degree and non-uniform are features of NURBS, whichwould make good additions to subdivision surfaces. Our works are mainly onsome problems about the compatibility of CAD and subdivision surfaces.Firstly, this paper introduce Lane-Riesenfeld algorithm, then introduce anon-symmetric, non-uniform and arbitrary degree subdivision algorithm forB-splines, and a symmetric, non-uniform and arbitrary degree subdivision forB-splines. Based on these algorithms, we present a non-uniform subdivisionalgorithm which share similar properties with Lane-Riesenfeld refine andsmooth construction, has well compatibility with NURBS. The resultingsurface is tensor-product spline surface，can represent any degree NURBSpatch exactly. As to control mesh of arbitrary topology, this algorithm firstlytransform extraordinary faces into regular faces, then transform the meshesnear extraordinary point of valency3into polar configurations, we get a limitsurface which isC1at the extraordinary points, but with bounded curvature,can be represented as a collection of tensor-product splines for any givendeviation tolerance. If these extraordinary points could be covered by polarconfigurations, then apply the subdivision algorithm of this paper, we obtaintensor-product spline surface in everywhere. Compared with the algorithm thatwas proposed by Thomas J.Cashman and Malcolm A. Sabin, our algorithm ismore efficient and concisely.