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Fast evaluation of derivatives of Bézier curves

Filip Chudy, Paweł Woźny

TL;DR

The paper addresses accelerating the computation of derivatives for polynomial and rational Bézier curves beyond the classical de Casteljau method. It introduces geometric, linear-time schemes that reuse a shared evaluation framework to compute derivatives up to any order, achieving complexities on the order of $O(rnd)$ for polynomials and $O(rd(n+r))$ for rational curves. Five approaches for rational curves are analyzed, including accelerated Floater formulas and differential-recurrence schemes, with extensive numerical tests showing speedups and informing method choice. The results improve practicality for high-degree curves and high-order derivatives in computer-aided design, enabling faster curvature and torsion computations and curve-merging tasks. The paper also outlines future work on numerical error analysis and derivative-bounds for rational Bézier curves.

Abstract

New geometric methods for fast evaluation of derivatives of polynomial and rational Bézier curves are proposed. They apply an algorithm for evaluating polynomial or rational Bézier curves, which was recently given by the authors. Numerical tests show that the new approach is more efficient than the methods which use the famous de Casteljau algorithm. The algorithms work well even for high-order derivatives of rational Bézier curves of high degrees.

Fast evaluation of derivatives of Bézier curves

TL;DR

The paper addresses accelerating the computation of derivatives for polynomial and rational Bézier curves beyond the classical de Casteljau method. It introduces geometric, linear-time schemes that reuse a shared evaluation framework to compute derivatives up to any order, achieving complexities on the order of for polynomials and for rational curves. Five approaches for rational curves are analyzed, including accelerated Floater formulas and differential-recurrence schemes, with extensive numerical tests showing speedups and informing method choice. The results improve practicality for high-degree curves and high-order derivatives in computer-aided design, enabling faster curvature and torsion computations and curve-merging tasks. The paper also outlines future work on numerical error analysis and derivative-bounds for rational Bézier curves.

Abstract

New geometric methods for fast evaluation of derivatives of polynomial and rational Bézier curves are proposed. They apply an algorithm for evaluating polynomial or rational Bézier curves, which was recently given by the authors. Numerical tests show that the new approach is more efficient than the methods which use the famous de Casteljau algorithm. The algorithms work well even for high-order derivatives of rational Bézier curves of high degrees.
Paper Structure (13 sections, 1 theorem, 62 equations, 3 figures, 8 tables, 2 algorithms)

This paper contains 13 sections, 1 theorem, 62 equations, 3 figures, 8 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Derivatives of polynomial Bézier curves
  3. New method
  4. Modification of the new method
  5. Numerical tests
  6. Derivatives of rational Bézier curves
  7. First method: accelerating Floater's formulas for the first and the second derivative
  8. Second method: using the scheme (1.9) to compute the derivative of any order
  9. Derivatives of the first and the second order
  10. Higher order derivatives
  11. Third method: using Eq. (3.1) to compute derivatives of any order
  12. Numerical tests
  13. Conclusions

Key Result

Theorem 3.1

Let $\hbox{R}_n : [0, 1]\rightarrow\mathbb{E}^d$ be a rational Bézier curve of degree $n$ defined in E:RationalBezierCurve. Then and where the numbers $\omega^{(i)}_k$ and points $\hbox{W}^{(i)}_k$ are computed by the rational de Casteljau algorithm (cf. E:RatdeCastel).

Figures (3)

  • Figure 1.1: A computation of a point on a planar polynomial Bézier curve of degree $n=5$ using the new method. The method computes much fewer intermediate points compared to the de Casteljau algorithm. Image taken from WCh2020.
  • Figure 3.1: The evaluation of the intermediate weights necessary for the base Floater's algorithm $(n=5)$. The arrows denote the steps of the rational de Casteljau algorithm. The same scheme applies for the necessary intermediate points $\hbox{W}^{(i)}_k$.
  • Figure 3.2: The evaluation of the intermediate weights necessary for the accelerated Floater's algorithm $(n=5)$. The one-headed arrows denote the steps of the rational de Casteljau algorithm, while the two-headed arrows represent the linear-time geometric method WCh2020. The same scheme applies for the necessary intermediate points $\hbox{W}^{(i)}_k$ (cf. Figure \ref{['F:Figure2']}).

Theorems & Definitions (8)

  • Remark 1.1
  • Remark 2.1
  • Example 2.2
  • Example 2.3
  • Theorem 3.1: Floater91
  • Example 3.2
  • Example 3.3
  • Example 3.4