To compute parametric representations for non-parametric algebraic intersection curves requires approximation. In this paper, the authors present algorithms to construct a piecewise rational \(B\)-spline approximation to a space curve, which comes from the intersection of two implicitly defined surfaces, or the intersection of two parametric defined surfaces. At singular points, the authors use the classical Newton power series factorization to determine the distinct branches of the space intersection curve. In addition to singular points, the authors obtain an adaptive selection of regular points about which the curve approximation yields a small number of curve segments yet achieves \(C^k\)-continuity between segments. Details of the implementation of these algorithm and approximation error bounds are also provided.