On the derivatives of rational Bézier curves
Mao Shi
TL;DR
This work addresses the challenge of computing higher-order derivatives for rational Bézier curves, which deteriorates when the derivative order $k$ approaches or exceeds the degree $n$. It introduces a recursive framework that expresses the $k$-th derivative as a $2^k n$-degree rational Bézier curve, with explicit update rules for the weights $\omega^{[k]}$ and corrected control quantities $\boldsymbol{P}^{[k]}$, starting from $\omega^{[0]}=\omega$ and $\boldsymbol{P}^{[0]}=\omega\boldsymbol{r}$. The authors also derive closed-form endpoint derivatives at $t=0$ and $t=1$, establish symmetry properties, and provide a practical derivative bound via Kuang’s inequality, supported by detailed $k=1,2,3$ examples. The approach enables robust symbolic and numerical manipulation of rational Bézier derivatives and can be extended to rational Bézier surfaces, improving endpoint behavior and derivative analysis in CAGD.
Abstract
By studying the existing higher order derivation formulas of rational Bézier curves, we find that they fail when the order of the derivative exceeds the degree of the curves. In this paper, we present a new derivation formula for rational Bézier curves that overcomes this drawback and show that the $k$th degree derivative of a $n$th degree rational Bézier curve can be written in terms of a $(2^kn)$th degree rational Bézier curve.we also consider the properties of the endpoints and the bounds of the derivatives.