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| Format: | Preprint |
| Published: |
2025
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| Online Access: | https://arxiv.org/abs/2510.17300 |
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| _version_ | 1866918362626392064 |
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| author | Shi, Mao |
| author_facet | Shi, Mao |
| contents | 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 $\sup_{t\in[0,1]}\|\mathbf r'(t)\|$. 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_17300 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Counterexamples to a Conjecture on First Derivative Bounds of Rational Bézier Curves Shi, Mao Numerical Analysis 65D17 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 $\sup_{t\in[0,1]}\|\mathbf r'(t)\|$. 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. |
| title | Counterexamples to a Conjecture on First Derivative Bounds of Rational Bézier Curves |
| topic | Numerical Analysis 65D17 |
| url | https://arxiv.org/abs/2510.17300 |