  Given a family of rational curves depending on a real parameter, defined by
its parametric equations, we provide an algorithm to compute a finite partition
of the parameter space (${\Bbb R}$, in general) so that the shape of the family
stays invariant along each element of the partition. So, from this partition
the topology types in the family can be determined. The algorithm is based on a
geometric interpretation of previous work (\cite{JGRS}) for the implicit case.
However, in our case the algorithm works directly with the parametrization of
the family, and the implicit equation does not need to be computed. Timings
comparing the algorithm in the implicit and the parametric cases are given;
these timings show that the parametric algorithm developed here provides in
general better results than the known algorithm for the implicit case.
