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

In this paper, an algorithm to determine the topology of an algebraic space curve and to compute a certified G rational parametric approximation of the algebraic space curve is given by extending the local generic position method for solving zero dimensional polynomial equation systems to the case of dimension one. By certified, we mean the approximation curve and the original curve have the same topology and their Hausdorff distance is smaller than a given precision. The main advantage of the algorithm, inherited from the local generic method, is that the topology computation and approximation for a space curve are directly reduced to the same tasks for two plane curves. In particular, the error bound of the approximation space curve is obtained from the error bounds of the approximation plane curves explicitly. We also analyze the complexity of computing the topology of an algebraic space curve. Nontrivial examples are used to show the effectivity of the method.