New minimal bounds for the derivatives of rational Bézier paths and rational rectangular Bézier surfaces

New minimal bounds are derived for the magnitudes of the derivatives of the rational Bézier paths and the rational rectangular Bézier surface patches of arbitrary degree, which improve previous work of this type in many cases. Moreover, our new bounds are explicitly given by simple and closed-form expressions. An important advantage of the closed-form expressions is that they allow us to prove that our bounds are sharp under certain well- defined conditions. Some numerical examples, highlighting the potential of the new bounds in providing improved estimates, are given in an appendix.