Discrete B-splines and subdivision techniques in computer-aided geometric design and computer graphics

Abstract

The relevant theory of discrete B-splines with associated new algorithms is extended to provide a framework for understanding and implementing general subdivision schemes for nonuniform B-splines. The new derived polygon corresponding to an arbitrary refinement of the knot vector for an existing B-spline curve, including multiplicities, is shown to be formed by successive evaluations of the discrete B-spline defined by the original vertices, the original knot vector, and the new refined knot vector. Existing subdivision algorithms can be seen as proper special cases. General subdivision has widespread applications in computer-aided geometric design, computer graphics, and numerical analysis. The new algorithms resulting from the new theory lead to a unification of the display model, the analysis model, and other needed models into a single geometric model from which other necessary models are easily derived. New sample algorithms for interference calculation, contouring, surface rendering, and other important calculations are presented.