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Rational complex Bezier curves

A. Canton, L. Fernandez-Jambrina, M. J. Vazquez-Gallo

TL;DR

The paper extends rational Bézier curves to the complex domain, introducing complex control points and weights and two distinct projective transformation groups (real and complex Möbius). It develops irreducibility tests via resultants in Bernstein form, proposes gcd-based factorisation in Bernstein basis, and shows how complex parametrisations can be converted to real ones or simplified by degree elevation. Through inversion-based constructions, it demonstrates degree reductions for classical curves (circles, cissoid, cardioid, lemniscate) and presents procedures to move between complex and real representations. The work provides both theoretical tools (resultants, gcd, inversion) and practical examples illustrating efficient curve design in CAD/CG applications. Overall, it broadens the design toolkit by embedding complex-valued weights and transformations into rational Bézier modeling, enabling new degrees of freedom and potential simplifications.

Abstract

In this paper we develop the formalism of rational complex Bezier curves. This framework is a simple extension of the CAD paradigm, since it describes arc of curves in terms of control polygons and weights, which are extended to complex values. One of the major advantages of this extension is that we may make use of two different groups of projective transformations. Besides the group of projective transformations of the real plane, we have the group of complex projective transformations. This allows us to apply useful transformations like the geometric inversion to curves in design. In addition to this, the use of the complex formulation allows to lower the degree of the curves in some cases. This can be checked using the resultant of two polynomials and provides a simple formula for determining whether a rational cubic curve is a conic or not. Examples of application of the formalism to classical curves are included.

Rational complex Bezier curves

TL;DR

The paper extends rational Bézier curves to the complex domain, introducing complex control points and weights and two distinct projective transformation groups (real and complex Möbius). It develops irreducibility tests via resultants in Bernstein form, proposes gcd-based factorisation in Bernstein basis, and shows how complex parametrisations can be converted to real ones or simplified by degree elevation. Through inversion-based constructions, it demonstrates degree reductions for classical curves (circles, cissoid, cardioid, lemniscate) and presents procedures to move between complex and real representations. The work provides both theoretical tools (resultants, gcd, inversion) and practical examples illustrating efficient curve design in CAD/CG applications. Overall, it broadens the design toolkit by embedding complex-valued weights and transformations into rational Bézier modeling, enabling new degrees of freedom and potential simplifications.

Abstract

In this paper we develop the formalism of rational complex Bezier curves. This framework is a simple extension of the CAD paradigm, since it describes arc of curves in terms of control polygons and weights, which are extended to complex values. One of the major advantages of this extension is that we may make use of two different groups of projective transformations. Besides the group of projective transformations of the real plane, we have the group of complex projective transformations. This allows us to apply useful transformations like the geometric inversion to curves in design. In addition to this, the use of the complex formulation allows to lower the degree of the curves in some cases. This can be checked using the resultant of two polynomials and provides a simple formula for determining whether a rational cubic curve is a conic or not. Examples of application of the formalism to classical curves are included.

Paper Structure

This paper contains 11 sections, 9 theorems, 37 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Rational complex Bézier curves
  3. Arcs of circles
  4. Degree elevation and irreducibility
  5. Factorisation
  6. From complex to real
  7. Some examples of curves
  8. Cissoid of Diocles
  9. Cardioid
  10. Lemniscate of Bernoulli
  11. Conclusions

Key Result

Theorem 1

Let $p(t)=\sum_{j=0}^{m}a_{j}t^{j}$, $q(t)=\sum_{j=0}^{n}b_{j}t^{j}$ be two polynomials with respective complex roots $\{x_{1},\ldots,x_{m}\}$, $\{y_{1},\ldots,y_{n}\}$, then

Figures (6)

  • Figure 1: Two arcs of circle from $z_{0}$ to $z_{1}$
  • Figure 2: Arc of circle with control polygon $\{1,1+i,i\}$ and weights $\{1,1,2\}$
  • Figure 3: Arc of parabola with control polygon $\{(1,0),(1,1),(0,1)\}$ seen as a rational cubic with control polygon $\{(1,0), (1, 4/5), (1/2, 1), (0, 1)\}$ and weights $\{2, 5/3, 4/3,1\}$
  • Figure 4: Arc of a cissoid of Diocles
  • Figure 5: Arc of a cardioid as inverse of a parabola
  • ...and 1 more figures

Theorems & Definitions (17)

  • Theorem 1
  • Corollary 1
  • Theorem 2
  • Corollary 2
  • Corollary 3
  • Example 1
  • Theorem 3
  • Example 2
  • Proposition 1
  • Lemma 1
  • ...and 7 more