Bi-Quadratic Final Polynomials

• Bi-Quadratic Final Polynomials are implicit equations representing biquadratic Bézier surfaces, derived via control point configurations using the Dixon resultant.
• They feature determinant-based constructions with degree 5 for triangles and degree 8 for quadrilaterals, ensuring efficient evaluation and robust geometric query capabilities.
• The approach leverages compact Cayley matrix formulations and precomputation strategies to optimize numerical performance in computer-aided geometric design.

A bi-quadratic final polynomial refers to the implicit equation generated in the implicitization of general biquadratic (order $2\times 2$) Bézier triangle and quadrilateral surface patches. This object, denoted as $F(x, y, z)$, is derived from the control points of the parametric surface and is used to represent the surface as the zero set of a single multivariate polynomial. The implicitization procedure employs the Dixon resultant, leading to an explicit determinant structure (the Cayley or Dixon matrix), and produces large-scale multivariate polynomials: degree 5 for the triangular case and degree 8 for the quadrilateral case. The expanded forms are multilinear in the control points, permitting efficient evaluation and providing a canonical algebraic framework for geometric queries and intersection computations (Borchardt et al., 2023).

1. Parametric Foundations

Biquadratic Bézier surfaces are specified by collections of control points in $\mathbb{R}^3$. For triangles (the $2 \times 2$ simplex case), six control points $$\{b_{200}, b_{020}, b_{002}, b_{110}, b_{101}, b_{011}\}$$ govern the surface

$$P(u,v) = b_{200}u^2 + b_{020}v^2 + b_{002}w^2 + 2b_{110}uv + 2b_{101}uw + 2b_{011}vw$$

where $u \geq 0$, $v \geq 0$, and $w = 1-u-v \geq 0$ are barycentric coordinates. For quadrilaterals (tensor-product $2 \times 2$ case), nine control points $p_{ij}$ with $0 \leq i, j \leq 2$ are used: $$P(u,v) = \sum_{i=0}^2\sum_{j=0}^2 B_i^2(u)B_j^2(v) p_{ij}$$ where $B_i^2$ are the quadratic Bernstein polynomials. The polynomials $f_x(u,v) = P_x(u,v)-x$, $f_y(u,v) = P_y(u,v)-y$, $f_z(u,v) = P_z(u,v)-z$ are constructed so that $(x, y, z)$ lies on the surface iff $f_x = f_y = f_z = 0$ for some $(u, v)$ in the domain (Borchardt et al., 2023).

2. Construction via the Dixon Resultant

Instead of performing sequential univariate eliminations, the Dixon resultant provides a compact determinant form to eliminate parameters $(u, v)$. This construction uses a $5\times5$ Cayley (Dixon) matrix for triangles and an $8\times8$ matrix for quadrilaterals. For the triangle, the Dixon resultant is given as: $$\delta(u,v,\alpha,\beta) = \frac{1}{(u-\alpha)(v-\beta)} \det\begin{bmatrix} f_x(u,v) & f_y(u,v) & f_z(u,v)\ f_x(u,\beta) & f_y(u,\beta) & f_z(u,\beta)\ f_x(\alpha,v) & f_y(\alpha,v) & f_z(\alpha,v)\ f_x(\alpha,\beta) & f_y(\alpha,\beta) & f_z(\alpha,\beta) \end{bmatrix}$$ and vanishes for all $(\alpha,\beta)$ if and only if $(x, y, z)$ lies on the Bézier patch. This is recast as a bilinear form in monomial bases and the vanishing of the determinant of the Cayley matrix yields $F(x, y, z) \equiv 0$ as the implicit equation (Borchardt et al., 2023).

3. Structure of the Implicit Polynomial

The implicit polynomial $F(x, y, z)$ resulting from the determinant is explicitly formulaic:

• For biquadratic Bézier triangles, $$F_3(x, y, z) = \sum_{i+j+k\leq5} C_{i,j,k}(b_{***}) x^i y^j z^k$$, with total degree 5 and 56 monomials.
• For biquadratic quadrilaterals, $$F_4(x, y, z) = \sum_{i+j+k\leq8} D_{i,j,k}(p_{ij,\ell}) x^i y^j z^k$$, with degree 8 and 165 monomials.

Each coefficient $C_{i,j,k}$ (triangle) or $D_{i,j,k}$ (quadrilateral) is a multilinear polynomial in the scalar control-point coordinates, and can be expressed as determinants or combinations of determinants of sub-arrays of the control points. For example, for the triangle: $$C_{500}(b) = \det \begin{bmatrix} b_{200,x} & b_{110,x} & b_{101,x} \ b_{200,y} & b_{110,y} & b_{101,y} \ b_{200,z} & b_{110,z} & b_{101,z} \ \end{bmatrix}$$

$$C_{410}(b) = 2\cdot\det \begin{bmatrix} b_{200,x} & b_{020,x} & b_{110,x} \ b_{200,y} & b_{020,y} & b_{110,y} \ b_{200,z} & b_{020,z} & b_{110,z} \ \end{bmatrix} - \det \begin{bmatrix} b_{020,x} & b_{110,x} & b_{101,x} \ b_{020,y} & b_{110,y} & b_{101,y} \ b_{020,z} & b_{110,z} & b_{101,z} \ \end{bmatrix}$$

This determinant-based expansion provides a canonical polynomial whose zero set defines the patch (Borchardt et al., 2023).

4. Evaluation and Implementation

Efficient implementation of $F(x, y, z)$ exploits the precomputation of all coefficients and Horner-style polynomial evaluation. For the triangle, code can be written as:

function F = evalF3_triangle(x, y, z, C) X = [1, x, x^2, x^3, x^4, x^5]; Y = [1, y, y^2, y^3, y^4, y^5]; Z = [1, z, z^2, z^3, z^4, z^5]; F = 0; for i = 0:5 for j = 0:(5-i) for k = 0:(5-i-j) F = F + C(i+1, j+1, k+1)*X(i+1)*Y(j+1)*Z(k+1); end end end end

Numeric evaluation of the fully expanded $F_3$ (triangle) requires $12,\!986$ multiplications and $269$ additions; coordinate normalization and optimized implementations (e.g., loop unrolling, FMA) can reduce this to $5,\!278$ multiplications for the triangle. The quadrilateral case, with degree 8, is structurally identical but with 165 terms and higher arithmetic complexity (Borchardt et al., 2023).

5. Computational Complexity and Optimization

The symbolic construction of the Cayley matrix has $O(n^3)$ complexity, where $n=3$ is the number of input polynomials. Determinant expansion for an $m \times m$ matrix ($m=5$ for triangles, $m=8$ for quadrilaterals) requires $O(m^3)$ symbolic operations. The final explicit polynomial for the triangle requires numeric evaluation of 56 terms (5,278 multiplications after normalization), while for the quadrilateral, 165 terms entail approximately $16$ million multiplications for the full expansion, or $\sim 4.7$ million for numeric Cayley determinant approximation.

Key optimizations include:

• Exploitation of the sparsity in the Cayley matrix (many zeros),
• Precomputation and reuse of recurrent minors or subresultants,
• Efficient polynomial evaluation schemes (FMA, loop unrolling).

Numeric evaluation of the determinant (without full expansion) for the triangle also requires approximately $5$ thousand floating-point operations, and yields results closely matching those from the explicit expanded formula (Borchardt et al., 2023).

6. Illustrative Example

For the case where control points for the Bézier triangle lie in a coordinate plane, the implicit polynomial simplifies dramatically. Take $b_{200} = (0,0,0)$, $b_{020} = (1,0,0)$, $b_{002} = (0,1,0)$, $b_{110} = (0,0,1)$, $b_{101} = (1,0,1)$, $b_{011} = (0,1,1)$. The corresponding polynomial,

$F(x, y, z) = (z-x)(z-y) = z^2 - (x+y)z + xy,$

illustrates factorization into two linear surfaces, confirming that the patch decomposes into two planar triangles. For these control points, the full Dixon resultant procedure yields the same polynomial directly from the determinant of the $5\times5$ Cayley matrix after clearing denominators and variable elimination (Borchardt et al., 2023).

7. Significance and Related Work

The use of the Dixon resultant to produce bi-quadratic final polynomials provides a principled mechanism for implicitization, allowing exact algebraic representations of parametric surfaces such as Bézier triangles and quadrilaterals, relevant in CAGD and geometric modeling. The procedure's computational tractability and its implicit polynomial output (canonical and multilinear in control points) facilitate robust surface-surface intersection, exact evaluation, and algebraic manipulation.

This approach, formalized by Borchardt & Kato (2024), generalizes classical constructions (Dixon, 1909; Sederberg et al., 1984) and harnesses modern computational resources for practical algebraic geometry in graphics and geometric computation. The low-dimensional Cayley determinant encodes the zero set of the parametric patch succinctly, and the resulting implicit equation is quickly evaluated either as an explicit polynomial or as a numerical determinant, providing near-identical precision at differing computational costs (Borchardt et al., 2023).