Rational Complex Bézier Curves

• Rational complex Bézier curves are parametric curves defined by complex control points and weights using a Bernstein basis, extending traditional curve design into the complex plane.
• Their formulation permits both real projective and Möbius transformations, enabling operations like geometric inversion and precise degree reduction.
• The approach offers practical CAD advantages by allowing concise representations of conic sections and other classic curves through computationally efficient methods.

A rational complex Bézier curve is a parametric curve defined by a finite set of complex control points and complex weights, with the curve coordinates given as a rational function in the Bernstein basis. By extending the traditional (real) CAD framework to the complex domain, rational complex Bézier curves inherit all projective and geometric features of rational Bézier curves, but also admit additional transformation and reduction operations via the use of complex projective (Möbius) transformations. This generalization provides enhanced modeling capabilities, allows geometric inversion and similar complex analytic operations, and gives algebraic criteria for degree reduction, notably distinguishing when a rational cubic is in fact a conic. The formalism directly encompasses both the algebra of curve parametrizations and the practical requirements of geometric design, providing new tools for analytical and computational manipulation of classic and novel curve families.

1. Formalism and Structure

A rational complex Bézier curve of degree $n$ is prescribed by $n+1$ complex control points $\{z_0, z_1, \ldots, z_n\} \subset \mathbb{C}$ and complex weights $$\{\omega_0, \omega_1, \ldots, \omega_n\} \subset \mathbb{C}$$. The curve is parametrized by

$$c(t) = \frac{\sum_{j=0}^{n} \omega_j z_j B_j^n(t)}{\sum_{j=0}^{n} \omega_j B_j^n(t)}, \quad t \in [0,1]$$

where $B_j^n(t) = \binom{n}{j} t^j (1-t)^{n-j}$ are the Bernstein basis polynomials. The control “polygon” $\{z_j\}$ determines the overall shape, with the complex weights influencing the curve’s internal proportions and orientations (for instance, a phase in the weight induces rotational effects in the tangent vector at the endpoints).

The projective lift in $\mathbb{C}^2$ is given by the polynomial curve

$$\vec{c}(t) = \sum_{j=0}^n (\omega_j, \omega_j z_j) B_j^n(t)$$

which enables coordinate-free manipulation under projective transformations.

2. Transformational Properties and Groups

The complex extension introduces two independent groups of projective transformations:

• Real projective group $\mathrm{PGL}_3(\mathbb{R})$, corresponding to standard operations in geometric design and classical CAD.
• Complex projective group $\mathrm{PGL}_2(\mathbb{C})$ (Möbius transformations), acting by $f(z) = (c + d z)/(a + b z)$ with $ad - bc \neq 0$.

These two groups are not subgroups of each other, expanding the set of admissible curve automorphisms. Crucially, the Möbius group includes geometric inversion ($f(z) = 1/z$), providing construction techniques unavailable in the real setting. When applied to a rational complex Bézier curve, inversion transforms control points as $z_j \mapsto 1/z_j$ and weights as $\omega_j \mapsto \omega_j z_j$, yielding new rational complex curves and facilitating the generation of circular arcs or other inverted geometries, such as the transformation of a line segment into an arc.

3. Degree Reduction and Polynomial Resultants

The use of complex coefficients often enables formal degree reduction of the rational Bézier parametrization:

• Degree Reduction: If numerator and denominator polynomials $p(t)$ and $q(t)$ (in either monomial or Bernstein basis) have a non-trivial common factor, the rational curve parametrization can be reduced by dividing by this factor, thereby lowering the degree.
• Sylvester Resultant: Reducibility is checked via the Sylvester resultant $R(p, q)$; in the Bernstein basis, this is constructed using “reduced coefficients” to absorb binomial weights. If $R(p, q) = 0$, cancellation and degree lowering are possible.

For rational cubics (degree three), the paper establishes a determinantal criterion for when such a parametrization represents an arc of a conic; the parametrization is a conic precisely when a specific $6 \times 6$ determinant (involving the weighted control points and Bernstein coefficients) vanishes. Explicitly, for control polygon $\{z_0, z_1, z_2, z_3\}$ and weights $\{\omega_0, \omega_1, \omega_2, \omega_3\}$,

$$\begin{vmatrix} \omega_0 z_0 & 3\omega_1 z_1 & 3\omega_2 z_2 & \omega_3 z_3 & 0 & 0 \ 0 & \omega_0 z_0 & 3\omega_1 z_1 & 3\omega_2 z_2 & \omega_3 z_3 & 0 \ 0 & 0 & \omega_0 z_0 & 3\omega_1 z_1 & 3\omega_2 z_2 & \omega_3 z_3 \ \omega_0 & 3\omega_1 & 3\omega_2 & \omega_3 & 0 & 0 \ 0 & \omega_0 & 3\omega_1 & 3\omega_2 & \omega_3 & 0 \ 0 & 0 & \omega_0 & 3\omega_1 & 3\omega_2 & \omega_3 \end{vmatrix} = 0$$

This provides a direct algebraic route to test whether a given rational cubic parametrization is in fact reducible to a conic section.

4. Practical Implications and Geometric Design Applications

• Classical Curve Parametrizations: Many classical plane curves admit succinct rational complex Bézier parametrizations of lower degree than is possible in the real case. For example, a circular arc can be represented as an affine segment in the complex plane by choosing weights $\omega_1/\omega_0 = e^{i\alpha}$, resulting in

$$c(t) = \frac{z_0(1-t) + e^{i\alpha} z_1 t}{(1-t) + e^{i\alpha} t}$$

which yields an arc that, in the real CAD paradigm, would require a quadratic (degree 2) representation.

• Projective Invariance: The rational complex Bézier formalism ensures that projective or Möbius transformation of the control polygon yields the same projective transformation of the curve, enabling seamless geometric operations (e.g., inversion, translation, rotation, dilation, and their complex analogues).
• Operational Manipulations: Degree elevation still applies as in the real case—multiplying numerator and denominator by a (possibly complex) linear factor raises the degree, preserving the underlying curve. Polynomial division (including Euclid’s algorithm in the Bernstein basis) supports algebraic simplification and degree reduction.
• CAD Implementation: Since the formalism retains the control polygon/weight paradigm, integration into existing CAD environments is operationally transparent, and compatibility with geometric design workflows is preserved.

5. Advanced Examples and Further Operations

• Constructive Examples: The formalism is utilized to parametrize:
  • The cissoid of Diocles, cardioid, and lemniscate of Bernoulli by complex rational Bézier formulas, usually after applying inversion and checking for degree reduction.
  • Classic geometric constructions such as the doubling of the cube, by unfolding conics into cissoids via Möbius maps.
• Algebraic Structure: The results on resultants and determinantal conditions underscore the deep interplay between algebraic and geometric properties in the rational complex domain, enabling symbolic simplification (reduction of parametrization degree) and explicit geometric classification (conic detection).
• Transformational Flexibility: The possibility of switching between the two projective transformation groups for curve manipulation represents a substantive generalization of the classic CAD toolkit, notably expanding the designer’s arsenal in both theoretical and applied contexts.

6. Summary Table

Feature	Real Rational Bézier	Rational Complex Bézier
Control points	$\mathbb{R}$ or $\mathbb{R}^2$	$\mathbb{C}$
Weights	$\mathbb{R}$	$\mathbb{C}$
Transformations	Real projective $\mathrm{PGL}_3(\mathbb{R})$	Real or complex projective ($\mathrm{PGL}_3(\mathbb{R})$ or Möbius $\mathrm{PGL}_2(\mathbb{C})$)
Inversion	Not Möbius in general	Möbius inversion (e.g., $z \mapsto 1/z$)
Degree reduction criterion	Sylvester resultant (real)	Sylvester resultant (complex), determinant vanishing for conic test
Classic curves (e.g., circle arc)	Degree 2	May admit degree 1 with complex weights

7. Conclusion

The rational complex Bézier curve formalism rigorously extends the foundational constructs of computer-aided geometric design to the complex domain, maintaining projective and geometric properties, while introducing significant new flexibility for transformations and degree minimization. The algebraic tools (Sylvester resultant, determinant criteria) provide explicit computational mechanisms for detecting reducibility and conic sections, while the connection to Möbius geometry unlocks deep integration with classical transformations, such as inversion. Applications span from succinct parametrization of classic curves to new methods for curve manipulation in complexified CAD settings, offering both theoretical insight and practical design power for advanced geometrical workflows (Canton et al., 31 Jul 2025).