  In this paper, an algorithm to compute a certified $G^1$ rational parametric
approximation for algebraic space curves 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 Hausdauff
distance is smaller than a given precision. Thus, the method also gives a new
algorithm to compute the topology for space algebraic curves. The main
advantage of the algorithm, inhering from the local generic method, is that
topology computation and approximation for a space curve is 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. Nontrivial examples are used to show the
effectivity of the method.
