Counterexamples to a Conjecture on First Derivative Bounds of Rational Bézier Curves

Abstract

In this paper we present an explicit counterexample of degree $n=7$, which shows that the conjecture proposed by Li et al. \cite{Li2013} regarding the first derivative bounds for rational Bézier curves is generally false. We further derive an explicit rational Bézier representation of the first derivative and propose a degree-elevation based computable upper bound for ${\sup }_{t\in [0,1]}\Vert {\mathbf{r}}^{\prime }(t)\Vert$. The bound is valid for any finite elevation order and converges to the true supremum as the elevation degree tends to infinity. An \emph{a priori} tolerance-driven rule is provided to determine a sufficient elevation degree, and the computational complexity of the proposed procedure is analyzed. Numerical experiments validate the counterexample and demonstrate the accuracy and efficiency of the new upper bound across a range of degrees and weight patterns.