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Counterexamples to a conjecture on first derivative bounds of rational Bézier curves

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  • Shi, Mao

Abstract

In this paper we present an explicit counterexample of degree n=7, which shows that the conjecture proposed by Li et al[1]. 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 supt∈[0,1]∥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 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.

Suggested Citation

  • Shi, Mao, 2026. "Counterexamples to a conjecture on first derivative bounds of rational Bézier curves," Applied Mathematics and Computation, Elsevier, vol. 528(C).
    • Handle: RePEc:eee:apmaco:v:528:y:2026:i:c:s0096300326002006
    DOI: 10.1016/j.amc.2026.130148
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