An application of Groebner bases to planarity of intersection of surfaces

In this paper we use Groebner bases theory in order to determine planarity of intersections of two algebraic surfaces in ${\bf R}^3$. We specially considered plane sections of certain type of conoid which has a cubic egg curve as one of the directrices. The paper investigates a possibility of conic plane sections of this type of conoid.

💡 Research Summary

The paper presents a systematic algebraic method for deciding whether the intersection curve of two algebraic surfaces in three‑dimensional Euclidean space lies entirely in a given plane. The authors base their approach on Gröbner‑basis theory, exploiting the fact that the ideal generated by the two surface equations (F(x,y,z)=0) and (G(x,y,z)=0) encodes all algebraic relations among the coordinates of points on the intersection. By computing a Gröbner basis (\mathcal{G}) of the ideal (\langle F,G\rangle) with respect to a lexicographic monomial order, one obtains a canonical generating set. The key observation is that a plane (P: ax+by+cz+d=0) contains the whole intersection if and only if the linear polynomial defining (P) belongs to the ideal (\langle F,G\rangle). In practice this is verified by forming the triple set ({F,G,P}) and recomputing a Gröbner basis; if the plane equation appears unchanged in the basis, the inclusion holds, otherwise it does not. This “Gröbner‑basis inclusion test” provides an exact, algorithmic criterion that avoids numerical approximation or geometric heuristics.

To illustrate the method, the authors focus on a special class of conoids whose directrix is a cubic “egg curve” given parametrically by (\gamma(t)=(t,;t^{3}+t,;0)). The conoid is described by a parametric representation (\mathbf{r}(u,v)=\mathbf{c}(u)+v,\mathbf{d}(u)) where (\mathbf{c}(u)=\gamma(u)) and (\mathbf{d}(u)) is a fixed direction vector (or a vector field orthogonal to the directrix). Eliminating the parameters yields two implicit polynomial equations (F_S(x,y,z)=0) and (F_T(x,y,z)=0) of degree three or higher. These equations define the surface (S) in (\mathbb{R}^3).

The authors then examine the intersection of (S) with a wide family of planes. For each candidate plane they compute the Gröbner basis of ({F_S,F_T,P}). In the majority of cases the plane polynomial does not survive in the basis, indicating that the intersection curve (C=S\cap P) is a higher‑degree algebraic curve (typically cubic or quartic) and therefore not a conic. However, for certain highly symmetric planes—specifically those parallel to the (x)-axis, orthogonal to the (z)-axis, or otherwise aligned with the geometry of the directrix—the plane equation does appear in the Gröbner basis. In these exceptional configurations the intersection reduces to a second‑degree curve (ellipse, parabola, or hyperbola). The paper thus demonstrates that, despite the presence of a cubic directrix, conic plane sections are possible only under very restrictive planar orientations.

Beyond the concrete example, the paper discusses computational aspects. Gröbner‑basis calculations become increasingly demanding as the degree of the defining polynomials grows, but modern computer algebra systems (Mathematica, Maple, Singular) implement optimized algorithms (F4/F5) that can handle the degree‑3–4 systems involved here within seconds on a standard workstation. The authors also explore the impact of variable ordering on the size of the intermediate bases and provide practical recommendations for choosing a lexicographic order that minimizes computational overhead.

In conclusion, the study establishes Gröbner‑basis techniques as a powerful, exact tool for testing planarity of surface intersections. It reveals that for a conoid with a cubic egg‑curve directrix, most planar cuts yield non‑conic algebraic curves, and only planes satisfying particular geometric alignments produce genuine conic sections. The methodology is general and can be extended to surfaces with higher‑degree directrices, multiple directrices, or even to non‑polynomial (trigonometric) surfaces, opening avenues for further research in algebraic geometry, computer‑aided design, and the analysis of ruled surfaces.