On the Topology of the Intersection Curve of Two Real Parameterized Algebraic Surfaces

Abstract

Let $\mathcal {S}_{\Gamma } : = {\operatorname {Im}} (\Gamma ) = \Gamma (\mathbb {R}^2)$ and $\mathcal {S}_{\varLambda } : = {\operatorname {Im}} (\varLambda ) = \varLambda (\mathbb {R}^2)$ two real algebraic parameterized surfaces defined by two integer polynomials maps: $\Gamma : \mathbb {R}^2 \longrightarrow \mathbb {R}^3$ and $\varLambda : \mathbb {R}^2 \longrightarrow \mathbb {R}^3$. We assume that the Zariski closures of $\mathcal {S}_{\Gamma }$ and $\mathcal {S}_{\varLambda }$ define a complete intersection and consider $\mathcal {C} (\mathbb {R}^3) := \mathcal {S}_{\Gamma } \cap \mathcal {S}_{\varLambda }$ the real algebraic intersection curve of the surfaces $\mathcal {S}_{\Gamma }$ and $\mathcal {S}_{\varLambda }$.

We give a method to compute the topology of $\mathcal {C} (\mathbb {R}^3)$ in terms of a simple straight-line space graph $\mathcal {G}$. The approach we introduce is based on the computation of a “well refined” topology of $\mathcal {C}_{s, t} (\mathbb {R}^2) := \Gamma ^{- 1} (\mathcal {C} (\mathbb {R}^3))$ the real algebraic curve preimage of $\mathcal {C} (\mathbb {R}^3)$ in the (s, t)-parametric real plane. The topology of $\mathcal {C}_{s, t} (\mathbb {R}^2)$ is computed in terms of a simple straight-line planar graph $\mathcal {D}= (\mathcal {V}, \mathcal {E})$ described by a set of vertexes $\mathcal {V}$ and a set of edges $\mathcal {E}$. The graph $\mathcal {D}= (\mathcal {V}, \mathcal {E})$ is said a “well refined” topology of ${\mathcal {C}_{s, t}} (\mathbb {R}^2)$ when it satisfies the remarkable property that, its image by Γ, the space graph $\mathcal {G}: = (\Gamma (\mathcal {V}), \Gamma (\mathcal {E}))$, with $\Gamma (\mathcal {V}) : = \{\Gamma (v) |v \in \mathcal {V}\}$ and $\Gamma (\mathcal {E}) : = \{(u, v) \in \Gamma (\mathcal {V}) \times \Gamma (\mathcal {V}) | (\Gamma ^{- 1} (u), \Gamma ^{- 1} (v)) \in \mathcal {E}\}$, is isotopic to the space curve ${\mathcal {C}}$(${\mathbb {R}}$ ³) . The keystone of our approach is the analysis and treatment of the thorny problems of unknotting of the space curve ${\mathcal {C}}$(${\mathbb {R}}$ ³) and disappearance of some of its singularities when one try to get its topology by computing the image by Γ of a planar graph isotopic to $\mathcal {C}_{s, t} (\mathbb {R}^2)$. We also introduce a specific process which reduce the problem of elimination to linear algebra computations. This new elimination algorithm help us efficiently compute the implicit equation of the plane curve $\mathcal {C}_{s, t} (\mathbb {R}^2)$.

Compare to existing methods, our approach uses symbolic-numeric computations and leads to a certified and general algorithm that compute the topology of the intersection curve of two surfaces parameterized by two arbitrary polynomials maps.

This work was completed with the support of our TE X-pert.