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In this paper, an algorithm to compute a certified $G^1$ rational parametricapproximation for algebraic space curves is given by extending the localgeneric position method for solving zero dimensional polynomial equationsystems to the case of dimension one. By certified, we mean the approximationcurve and the original curve have the same topology and their Hausdauffdistance is smaller than a given precision. Thus, the method also gives a newalgorithm to compute the topology for space algebraic curves. The mainadvantage of the algorithm, inhering from the local generic method, is thattopology computation and approximation for a space curve is directly reduced tothe same tasks for two plane curves. In particular, the error bound of theapproximation space curve is obtained from the error bounds of theapproximation plane curves explicitly. Nontrivial examples are used to show theeffectivity of the method.

Certified Rational Parametric Approximation of Real Algebraic Space Curves with Local Generic Position Method