Degree elevation operator and geometric construction of C-B-spline curves

Abstract

Unlike a Bezier curve, a spline curve is hard to be obtained through geometric corner cutting on control polygons because the degree elevation operator is difficult to be obtained and geometric convergence is hard to be achieved. In order to obtain geometric construction algorithm on C-B-splines, firstly we construct the degree elevation operator by using bi-order splines in this paper. Secondly we can obtain a control polygon sequence by degree elevation based on the degree elevation operator derived from a C-B-spline curve. Finally, we prove that this polygon sequence will converge to initial C-B-spline curve. This geometric construction algorithm possesses strong geometric intuition. It is also simple, stable and suitable for hardware to perform. This algorithm is important for CAD modeling systems, since many common engineering curves such as ellipse, helix, etc. can be represented explicitly by C-B-splines.