On the First Derivative Bounds for Rational Bézier Curves

Abstract

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 $\Vert {\mathbf{R}}^{\prime }(t)\Vert \le n\Omega \max \Vert {\Delta }_i\Vert$ holds for all degrees. We prove that the bound is indeed valid for $n\le 6$, 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.

21pages