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arXiv · 2607.10425

On the First Derivative Bounds for Rational B\'ezier Curves

Mao Shi
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Abstract

In this paper we investigate sharp upper bounds for the first derivative of rational B\'ezier curves. A long-standing conjecture posited that the linear bound $\|\mathbf{R}'(t)\| \le n\Omega \max\|\Delta_i\|$ holds for all degrees. We prove that the bound is indeed valid for $n \leq 6$, thus resolving the last open low-degree case. The problem is reformulated as maximizing a variance-like function over a compact box. Using a block argument we show that optima can only appear on one-dimensional faces, reducing the task to a finite family of polynomial inequalities, which are verified exactly via real quantifier elimination. A notable practical feature is that the bound can be evaluated in linear time with respect to the degree, making it attractive for real-time geometric processing. The same structural analysis illustrates the failure for $n=7$ and outlines how the true worst-case constant can be computed.

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Mao Shi. 2026-07-11. On the First Derivative Bounds for Rational B\'ezier Curves. https://arxiv.org/abs/2607.10425

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