On the First Derivative Bounds for Rational Bézier Curves

Jul 16, 2026

Investigates sharp upper bounds for the first derivative of rational Bézier curves and resolves a long-standing conjecture. The linear inequality of the form ||R′(t)|| ≤ nΩ max||Δi|| is proved valid for degrees up to n ≤ 6, turning the task into maximizing a variance-like scalar function over a compact box of admissible adjacent weight ratios. A block/critical-point structure shows maxima occur only on low-dimensional faces, reducing the claim to finitely many polynomial inequalities verified by real quantifier elimination, and analyzing the failure at n = 7.

LanguageEnglish

CategoryResearch & Report

FormatPDF

Pages21

Reading Time53 min

Introduction

Main result for degree n ≤ 6

Reduction to semi-algebraic optimization

Quantifier elimination verification

Complexity of the bound

Theorem and supporting lemma

arXiv :2607 . 10425v1 [math .NA] 11 Jul 2026 Highlights On the First Derivative Bounds for Rational Bézier Curves Mao Shi • Rigorously proves a first derivative bound conjecture for rational Bézier curves of degree 6 . • Reduces the global optimization problem to the 1-dimensional skeleton of a compact box. • Employs exact symbolic computation (quantifier elimination) for a finite case verification. • Demonstrates the conjecture is false for degree 7 and provides a method for computing the true bound. • The verified first derivative bound is computable in O (n) time, suitable for real-time applications. On the First Derivative Bounds for Rational Bézier Curves Mao Shi School of Mathematics and Statistics, Shaanxi Normal University, Xi’an, China 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 ∥R′(t)∥ ≤ nΩ max ∥∆i ∥ holds for all degrees. We prove that the bound is indeed valid for n ≤ 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. Keywords: Rational Bézier curves, First derivative bounds, Semi-algebraic optimization, Quantifier elimination 1. Introduction Estimating the first derivative bounds of rational Bézier curves is a fundamental problem in Computer Aided Geometric Design (CAGD) . Let R (t) = P ωiBni(t)Pi 0 ≤ t ≤ 1 (1) P ωiBni(t) , , be a degree-n rational Bézier curve with control points Pi ∈ Rd and positive weights ωi , where Bni(t) =􀀐ni􀀑ti (1 − t)n−i are the Bernstein basis functions. Let ∆i = Pi+1 − Pi denote the control vector of the polygon. Floater [1] derived the first derivative inequality ∥R′ ∥ ≤ n 􀀐maxmin~~ω~~ω~~i~~i 􀀑 2 max ∥∆i ∥ . Selimović [2] improved this by using the maximal adjacent weight ratio Ω = 0i1 max ( ωi~~ ~~1 , ωi~~ ~~1 ) , (2) obtaining ∥R′ ∥ ≤ nΩn max ∥∆i ∥ . Based on the derivative formula of rational Bézier curves given by Sederberg and Wang [3], Zhang and Ma [4, 5] derived the tighter bound ∥R′(t)∥ ≤ nΩ max ∥∆i ∥ for n = 2 , 3 , 4. Using extensive numerical experiments, Li et al. [6] conjectured that this linear first derivative bound holds for all degrees n. However, Shi [7] constructed counterexamples showing the inequality fails for n ≥ 7. The cases n = 5 and n = 6 have remained open. In this paper we prove the conjecture for n = 6 . The cases n = 2 , 3 , 4 , 5 can be handled by the same method. Our approach is based on a reduction of the geometric inequality to a global optimization problem over a compact semi-algebraic set, namely the box of admissible adjacent weight ratios. We then show, via a structural analysis of the critical point equations, that the maximum of the resulting scalar function can only be attained on the low-dimensional boundary of this box—more precisely, on its vertices or one-dimensional edges. This reduction transforms the original analytic statement into a finite family of polynomial inequalities involving two real variables, u > 0 and Ω ≥ 1. Each of these conditions is a universally quantified formula of the form ∀ u > 0 , Ω ≥ 1 : Φ(u, Ω) ⇒ Ξ(u, Ω) ≥ 0, which belongs to the decidable theory of real closed fields. The method of quantifier elimination, a fundamental concept in mathematical logic (see, e.g., Rosen [8] or Shi [9] for an accessible introduction), was shown to be applicable to real closed fields by Tarski [10

FAQs

What conjecture is proved in the paper for rational Bézier curves?

It proves the linear first-derivative bound of the form ||R′(t)|| ≤ nΩ max ||Δi|| for degrees n ≤ 6, completing the last unresolved low-degree case.

How is the optimization problem reformulated to prove the bound?

The problem is reformulated as maximizing a variance-like scalar function over a compact box of admissible adjacent weight ratios, turning the analytic bound into polynomial inequality checks.

Why does the bound fail for degree n = 7, and what does the paper propose instead?

The structural analysis shows the inequality does not hold for n = 7 and outlines how to compute the true worst-case constant by the same kind of reduction.

Jul 16, 2026

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