The Improvement On The T-spline Local Refinement Algorithm

The de-facto technology standard for representing curved surfaces since the 1970s is non-uniform rational B-splines (NURBS) which is most widely used in engineering de-sign. The major strengths of NURBS are that they are convenient for freeform surface modeling, can exactly represent all quadric surfaces, and that there exist many efficient and numerically stable algorithms to generate NURBS objects. They also possess useful mathematical properties, such as the ability to be refined through knot insertion, Cp-l-continuity for degree p NURBS, and the variation diminishing and convex hull properties. NURBS are ubiquitous in Computer-Aided Design (CAD) systems, representing billions of dollars in development investment. The major deficiencies of NURBS are that gaps and overlaps at intersections of surfaces cannot be avoided, complicating mesh generation, and that they utilize a tensor product structure making the representation of detailed local features inefficient.In order to overcome the weaknesses of NURBS, T. W. Sederberg et al. proposed the concept of T-spline in 2003. T-splines correct the deficiencies of NURBS in that they permit local refinement and coarsening, and a solution to the gap/overlap problem. Subse-quently, in 2004, a very important core algorithm, the T-spline local refinement algorithm, was presented by T. W. Sederberg et al.. However, this algorithm is complicated, and is not convenient to obtain the location of the additional control points directly from the topology.In this thesis, an equivalent description using T-junction extension is given, which can simplify the process of judging whether an additional point should be inserted or not. This method can generate the final T-mesh faster. According to different cases of inserting edges, we discuss the improved algorithm in both simple case and general case, and then draw a complete conclusion.T-spline modeling technology has been applicated for several years. T-Splines for Rhino now is still the only commercial software which offers operations on surface using T-spline technology. It is a plug-in for Rhinoceros which is a NURBS-based 3D modeling tool. In this thesis, we also introduce T-Spline for Rhino briefly for verifying our algorithm.