Approximating uniform rational B-spline curves by polynomial B-spline curves

摘要

Approximation of rational B-spline curves by B-spline curves is an important issue in computer aided geometric design. This paper presents a method to approximate a uniform rational B-spline with B-spline curve sequence as follows. We first elevate the degree of the original rational B-spline curve and take the control points of the degree-elevated curve as new control points of the B-spline approximation curve. Next we take an extended knot vector of the original curve as a new knot vector of the approximation curve. This generates a B-spline approximation curve with the same degree as the degree-elevated curve. Based on the discrete B-spline and multiple products of B-spline functions, we finally prove that the derivatives of any given degree of the uniform B-spline approximation curve sequence converge uniformly to the corresponding derivatives of the original rational B-spline curve. This approximation method is very simple and guarantees the convergence of the approximation.