On the convergence of hybrid polynomial approximation to higher derivatives of rational curves

Abstract

In this paper, we derive the bounds on the magnitude of lth ( l = 2 , 3 ) order derivatives of rational Bézier curves, estimate the error, in the L ∞ norm sense, for the hybrid polynomial approximation of the lth ( l = 1 , 2 , 3 ) order derivatives of rational Bézier curves. We then prove that when the hybrid polynomial approximation converges to a given rational Bézier curve, the lth ( l = 1 , 2 , 3 ) derivatives of the hybrid polynomial approximation curve also uniformly converge to the corresponding derivatives of the rational curve. These results are useful for designing simpler algorithms for computing tangent vector, curvature vector and torsion vector of rational Bézier curves.