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