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Counterexamples to the conjecture of the upper bound of the derivative of a rational Bézier curve

Mao Shi

TL;DR

The paper addresses whether a universal upper bound for the first derivative of rational Bézier curves holds, by constructing counterexamples to the Li et al. conjecture and by developing a method to characterize the supremum of derivatives of all orders. It derives an explicit first-derivative representation through degree elevation and norm-based bounds, enabling computation of the derivative supremum and extension to higher-order derivatives (aligned with recent high-order formulas). A concrete $n=11$ counterexample demonstrates the conjecture's failure, and subsequent numerical experiments illustrate a practical framework for bounding derivatives and exploring the supremum across orders. The work informs stability considerations in geometric design and raises open questions about the conditions under which the conjecture may fail and the influence of weight convergence.

Abstract

In this paper, we present counterexamples to the upper bound of the first-order derivative of rational Bézier curves and further investigate the supremum of derivatives of all orders for such curves.

Counterexamples to the conjecture of the upper bound of the derivative of a rational Bézier curve

TL;DR

The paper addresses whether a universal upper bound for the first derivative of rational Bézier curves holds, by constructing counterexamples to the Li et al. conjecture and by developing a method to characterize the supremum of derivatives of all orders. It derives an explicit first-derivative representation through degree elevation and norm-based bounds, enabling computation of the derivative supremum and extension to higher-order derivatives (aligned with recent high-order formulas). A concrete counterexample demonstrates the conjecture's failure, and subsequent numerical experiments illustrate a practical framework for bounding derivatives and exploring the supremum across orders. The work informs stability considerations in geometric design and raises open questions about the conditions under which the conjecture may fail and the influence of weight convergence.

Abstract

In this paper, we present counterexamples to the upper bound of the first-order derivative of rational Bézier curves and further investigate the supremum of derivatives of all orders for such curves.
Paper Structure (5 sections, 2 theorems, 21 equations, 2 figures, 1 table)

This paper contains 5 sections, 2 theorems, 21 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. Counterexample
  3. Supremum
  4. Example
  5. Conclusion

Key Result

Theorem 1

The explicit expression for the first derivative of a rational Bézier curve is where the parameters are calculated as follows 1. Weights $\omega_{i,2n}^{[1]}$ ($i=0,\dots,2n$) 2. Intermediate control points $\boldsymbol{P}_{j,2n}^{[1]}$ ($j=0,\dots,2n-1$) and 3. Control points ${\bf{Q}}_{i,2n}^{[1]}$ ($i=0,\dots,2n$)

Figures (2)

  • Figure 1: The rational Bézier curve $r(t)$ ($n=11$).
  • Figure 2: The derivative curve $r'(t)$ and the reference line of the conjectured upper bound.

Theorems & Definitions (3)

  • Conjecture 1
  • Theorem 1
  • Theorem 2