Hybrid polynomial approximation to higher derivatives of rational curves

In this paper, we extend the results published in JCAM volume 214 pp. 163-174 in 2008. Based on the bound estimates of higher derivatives of both Bernstein basis functions and rational Bezier curves, we prove that for any given rational Bezier curve, if the convergence condition of the corresponding hybrid polynomial approximation is satisfied, then not only the l-th (l = 1, 2, 3) derivatives of its hybrid polynomial approximation curve uniformly converge to the corresponding derivatives of the rational Bezier curve, but also this conclusion is tenable in the case of any order derivative. This result can expand the area of applications of hybrid polynomial approximation to rational curves in geometric design and geometric computation. (C) 2011 Elsevier B.V. All rights reserved.