Parametrization of translational surfaces

Abstract: The algebraic translational surface is a typical modeling surface in computer aided design and architecture industry. In this paper, we give a necessary and sufficient condition for that algebraic surface having a standard parametric representation and our proof is constructive. If the given algebraic surface is translational, then we can compute a standard parametric representation for the surface.

• The paper establishes a constructive necessary and sufficient criterion to decide if an algebraic surface admits a translational parametrization.
• It introduces explicit algorithms using rational space curves and derivative conditions to compute proper reparametrizations.
• The approach has practical implications for CAD and geometric modeling by converting implicit representations to precise parametric forms.

Parametrization of Translational Surfaces

Introduction

The study of algebraic surfaces admitting explicit parametric representations is essential in geometric modeling, CAD, and related computational fields. Translational surfaces, defined by a sum of two space curve parametrizations, $$\mathcal{P}(t_1, t_2) = \mathcal{P}_1(t_1) + \mathcal{P}_2(t_2)$$, are of particular practical importance due to their simplicity and prevalence in industrial design and surface interpolation. This work addresses the characterization and constructive parametrization of algebraic translational surfaces, specifically determining necessary and sufficient algebraic conditions for when such a representation exists and presenting algorithms for computing the parametrization in the affirmative case.

Algebraic Formulation and Main Results

The paper studies varieties $\mathcal{V} \subset \mathbb{K}^3$ (over an algebraically closed field of characteristic zero) defined by an irreducible polynomial equation $f(x_1,x_2,x_3)=0$. The key question is to decide whether $\mathcal{V}$ admits a parametrization of the form $$\mathcal{P}(t_1, t_2) = \mathcal{P}_1(t_1) + \mathcal{P}_2(t_2)$$, with $$\mathcal{P}_i(t_i) \in \mathbb{K}(t_i)^3 \setminus \mathbb{K}^3$$, and, if so, to compute it explicitly. The paper specifically excludes cylindrical and planar surfaces, for which this question is trivial.

The central result is a constructive necessary and sufficient criterion for the existence of such a parametrization. The authors show that translationality is intimately connected to the existence of rational space curves defined by the intersection of the surface with a certain system of additional algebraic equations derived from $f$ and its partial derivatives.

Key Theorem: Constructive Criterion

Given an irreducible surface $f(x_1, x_2, x_3) = 0$, $\mathcal{V}$ is a translational surface if and only if:

1. There exists a vector $$(a_1,a_2,a_3)\in \mathbb{K}^3 \setminus \{(0,0,0)\}$$ such that the intersection of $f(x) = 0$ with $$g(x):=a_1 f_{x_1}(x) + a_2 f_{x_2}(x) + a_3 f_{x_3}(x)=0$$ defines a rational space curve $\mathcal{C}_1$, which can be rationally parametrized as $\mathcal{P}_1(t_1)$.
2. Considering the shifted surface polynomial $f(\mathcal{P}_1(t_1) + x) = 0$, the locus of $x$ for which this vanishes for all $t_1$ can be written as $\Psi(x, t_1)=0$ (modulo some extraneous factors). A rational space curve contained in the base locus $V(\Psi_0, \ldots, \Psi_n)$, for generic $t_1$, can be found and parametrized as $\mathcal{P}_2(t_2)$.

The ultimate parametrization is then $$\mathcal{P}(t_1, t_2) = \mathcal{P}_1(t_1) + \mathcal{P}_2(t_2)$$, provided both curves are non-linear and irreducible.

Methodological Contributions

Properness and Reparametrization

The work rigorously formalizes properness conditions: if the surface parametrization is proper, then so are $\mathcal{P}_1$ and $\mathcal{P}_2$. The converse does not necessarily hold, but the existence of proper reparametrizations is established using earlier results on the properness of rational curves.

Algorithmic Constructiveness

Unlike previous approaches limited to special classes of surfaces or reliant on ad hoc coordinates, the present method is constructive and canonical:

• The vector $(a_1,a_2,a_3)$ can be chosen as a non-trivial tangent direction for the sought curve $\mathcal{C}_2$, often conveniently picked from the derivative of $\mathcal{P}_2$ at a suitable base point.
• Parametric computation of $\mathcal{C}_1$ leverages standard algorithms for the intersection of algebraic varieties and the rational parametrization of space curves.
• For $\mathcal{C}_2$, the base locus equations reduce the computation to the intersection curves defined by two specialized equations $\Psi(x, s_1) = \Psi(x, s_2) = 0$ for generic $s_1, s_2$, further enhancing algorithmic tractability.

Examples

The procedure is demonstrated with explicit examples, including quadratic and higher degree surfaces, emphasizing the versatility of the approach. In all examples, strong claims are validated: the construction yields an explicit rational parametrization or certifies the non-existence of a translational structure. For instance, the method correctly identifies and constructs parametrizations for surfaces such as $x_3 + 5x_1^2 - 6x_1 x_2 + 2x_2^2 = 0$, providing explicit rational generators for both $\mathcal{C}_1$ and $\mathcal{C}_2$.

Implications and Discussion

Theoretical Implications

This work provides the first fully constructive birational classification of translational algebraic surfaces: the derived necessary and sufficient condition settles the algebraic question of when a variety admits a translational parametrization. By utilizing the partial derivatives of the defining polynomial, the approach is optimal in the sense that the surface's geometry is completely encoded in its differential invariants.

The paper also analyzes the space of admissible directions $(a_1,a_2,a_3)$, connecting the structure of the surface's conical symmetries to the parametrization problem. This clarifies why only non-cylindrical, non-planar surfaces are genuinely non-trivial for the translational parametrization question.

Practical Relevance

The results have direct applications for geometric modeling and implicit/parametric data conversion in CAD. The algorithmic procedures permit the recovery of standard parametric representations from implicit descriptions, enabling efficient rendering, intersection calculations, and further symbolic manipulation. The reduction to the intersection of varieties and the specialization of parameter values make the method computationally practical.

Numerically, the paper demonstrates that in all worked cases, rational parametrizations are derived without recourse to irrational functions, which is crucial for exact geometric computation.

Limitations and Extensions

The method is inherently limited to rationally parametrizable surfaces; transcendental structures or highly singular surfaces fall outside the scope. Furthermore, the uniqueness (up to birational transformations) of the parametrization is not addressed—the space of admissible parametrizations is typically infinite.

Potential extensions include adaptations to fields of positive characteristic and further study of the moduli space of all possible translational parametrizations for a given surface. Links with the theory of Darboux and Bäcklund transformations in surface geometry may also yield new insights.

Conclusion

This paper establishes a definitive, constructive criterion and algorithm for the existence and computation of parametric representations of algebraic translational surfaces. The results provide both theoretical completeness and computational applicability. They enable efficient conversion between implicit and parametric models, critical for geometric design and industrial applications. Further work may extend these methods to broader classes of surfaces and explore the algebraic-geometric landscape of translational parametrizations in higher dimensions.