AI 中文总结
研究有理贝塞尔曲线一阶导数的精确上界，通过将问题转化为在紧凑盒子上最大化类似方差的函数，证明$n \leq 6$时线性界有效，解决低次开放情况，且该界可线性时间求值，还说明了$n = 7$时界不成立及计算最坏情况常数的方法。
AI 中文摘要
本文研究有理贝塞尔曲线一阶导数的精确上界。长期存在的一个猜想是线性界$\|\mathbf{R}'(t)\| \le n\Omega \max\|\Delta_i\|$对所有次数都成立。我们证明该界对于$n \leq 6$确实有效，从而解决了最后一个低次开放情况。问题被重新表述为在一个紧凑盒子上最大化一个类似方差的函数。通过块论证表明最优值只能出现在一维面上，将任务简化为有限的多项式不等式族，通过实量词消去精确验证。一个显著的实际特点是该界可以关于次数在线性时间内求值，对实时几何处理有吸引力。相同的结构分析说明了$n = 7$时该界不成立，并概述了如何计算真正的最坏情况常数。
英文摘要
In this paper we investigate sharp upper bounds for the first derivative of rational Bézier curves. A long-standing conjecture posited that the linear bound $\|\mathbf{R}'(t)\| \le nΩ\max\|Δ_i\|$ 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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