From NURBS to C-NURBS: I — C-NURBS Curves and Their Properties

This paper develops a new free-form model, called c-NURBS, which is a general model of NURBS. A c-NURBS curve or surface is the projection of a 6D B-spline curve from a 6D homogeneous space H⁶ into a 3D space R³. The construction procedure of a c-NURBS curve or surface is that using cubic curves or bicubic patches repeatedly and piecewisely interpolates the given control points. The distinct properties of c-NURBS include independent weight modification, super-convexity, strong c-convex hull, and hidden degrees and control points. These properties greatly enhance the shape control and modification capability. All techniques developed for NURBS, such as the de Boor-Cox algorithm, knot insertion, and degree elevation and reduction, can be applied to c-NURBS. The implementation of c-NURBS requires little improvement on the CAD/CAM systems based on NURBS.