Counterexamples to a Conjecture on First Derivative Bounds of Rational B\'ezier Curves

TL;DR

This paper provides a counterexample to a conjecture on the first derivative bounds of rational Bézier curves, introduces a new computable upper bound, and validates it through numerical experiments.

Contribution

It presents the first explicit counterexample to the conjecture and proposes a degree-elevation based method for an accurate upper bound on the derivative norm.

Findings

Counterexample invalidates the conjecture.

New upper bound converges to the true supremum.

Method is efficient and accurate across various degrees.

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 B\'ezier curves is generally false. We further derive an explicit rational B\'ezier representation of the first derivative and propose a degree-elevation based computable upper bound for $\sup_{t\in[0,1]}\|\mathbf r'(t)\|$. 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.