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Application of a metric for complex polynomials to bounded modification of planar Pythagorean-hodograph curves

Rida T. Farouki, Marjeta Knez, Vito Vitrih, Emil Žagar

TL;DR

The paper presents a metric-based framework for complex polynomial representations of planar curves to quantify perturbations while preserving planar Pythagorean-hodograph (PH) structure. By treating PH pre-image polynomials as complex functions and using inner products, norms, and a distance on polynomials, the authors define bounded modifications of the pre-image ${\bf w}(t)$ that keep the resulting PH curve valid. They develop both Bernstein and Legendre representations to bound perturbations and derive methods to preserve end points and end tangents (in canonical form) or to adjust arc length by a prescribed amount, including practical algorithms and illustrative examples. The approach generalizes to spatial PH curves via quaternions and hints at extensions to parametric surfaces, offering a practical tool for real-time, PH-safe shape design with controlled geometric changes.

Abstract

By interpreting planar polynomial curves as complex-valued functions of a real parameter, an inner product, norm, metric function, and the notion of orthogonality may be defined for such curves. This approach is applied to the complex pre-image polynomials that generate planar Pythagorean-hodograph (PH) curves, to facilitate the implementation of bounded modifications of them that preserve their PH nature. The problems of bounded modifications under the constraint of fixed curve end points and end tangent directions, and of increasing the arc length of a PH curve by a prescribed amount, are also addressed.

Application of a metric for complex polynomials to bounded modification of planar Pythagorean-hodograph curves

TL;DR

The paper presents a metric-based framework for complex polynomial representations of planar curves to quantify perturbations while preserving planar Pythagorean-hodograph (PH) structure. By treating PH pre-image polynomials as complex functions and using inner products, norms, and a distance on polynomials, the authors define bounded modifications of the pre-image that keep the resulting PH curve valid. They develop both Bernstein and Legendre representations to bound perturbations and derive methods to preserve end points and end tangents (in canonical form) or to adjust arc length by a prescribed amount, including practical algorithms and illustrative examples. The approach generalizes to spatial PH curves via quaternions and hints at extensions to parametric surfaces, offering a practical tool for real-time, PH-safe shape design with controlled geometric changes.

Abstract

By interpreting planar polynomial curves as complex-valued functions of a real parameter, an inner product, norm, metric function, and the notion of orthogonality may be defined for such curves. This approach is applied to the complex pre-image polynomials that generate planar Pythagorean-hodograph (PH) curves, to facilitate the implementation of bounded modifications of them that preserve their PH nature. The problems of bounded modifications under the constraint of fixed curve end points and end tangent directions, and of increasing the arc length of a PH curve by a prescribed amount, are also addressed.
Paper Structure (9 sections, 1 theorem, 96 equations, 8 figures)

This paper contains 9 sections, 1 theorem, 96 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Metric space of complex polynomials
  3. Orthogonal planar curves
  4. Planar Pythagorean-hodograph curves
  5. Perturbation of pre-image polynomials
  6. Preservation of curve end tangent directions
  7. Preservation of curve end points
  8. Modification of PH curve arc lengths
  9. Closure

Key Result

Lemma 1

For a polynomial $p(t)$ of degree $n$ expressed in the Legendre and Bernstein bases on $[\,0,1\,]$ as the coefficients $\boldsymbol{C}=(c_0,\ldots,c_n)^T$ and $\boldsymbol{D}= (d_0,\ldots,d_n)^T$ are related according to $\boldsymbol{D}=M_n\boldsymbol{C}$, where $M_n$ is the $(n+1)\times(n+1)$ matrix with elements whose inverse $M_n^{-1}$ has elements Note that the columns of the matrix $M_n$ a

Figures (8)

  • Figure 1: Left: the curve ${\bf r}(t)$ in Example \ref{['exm:orthogonal-1']} is indicated in black, and four curves ${\bf r}_\perp(t)$ orthogonal to it are shown in red, green, blue, and purple. Right: the graphs of $\hbox{Re}({\bf r}(t)\, {\bf r}_\perp(t))$ for these four curves.
  • Figure 2: The cubic PH curve ${\bf r}(t)$ (black) in Example \ref{['exm:orthogonal-2']}, together with the six PH curves ${\bf r}_{\perp}(t)$ orthogonal to it (shown in different colors) that possess the same start point $(0,0)$ and have the prescribed parametric speed $\sigma(t)$.
  • Figure 3: Left: The prescribed quintic PH curve (blue), with four instances modified using the Bernstein basis (different colors), whose pre--images satisfy $\|\delta{\bf w}\|=0.25$, as described in Example \ref{['exm:example1']}. Right: The envelope of the family of all perturbed curves with preserved end tangent directions for $r=0.25$.
  • Figure 4: Left: The prescribed quintic PH curve (blue) with four instances modified using the Legendre basis (different colors), whose pre--images satisfy $\|\delta{\bf w}\|=0.25$, as described in Example \ref{['exm:example1']}. Right: The envelope of the family of all perturbed curves with preserved end tangent directions for $\rho=0.25/\sqrt{3}$.
  • Figure 5: Left: The perturbed PH quintics (red and green) for $\|\delta {\bf w}\| = 0.1$, with the same end points as the quintic PH curve (blue) in Example \ref{['exm:example5']}, defined by $\varphi=0$. The envelope of all the solutions for $\varphi\in (-\,\pi,\pi\,]$ is also shown (light blue). Right: The modified PH quintic curves that preserve end points and end tangent directions, as described in Example \ref{['exm:tangents']}, as $\|\delta {\bf w}\|$ varies.
  • ...and 3 more figures

Theorems & Definitions (10)

  • Example 1
  • Example 2
  • Lemma 1
  • Example 3
  • Example 4
  • Example 5
  • Example 6
  • Remark 1
  • Example 7
  • Example 8