  Given the implicit equation $F(x,y,t,s)$ of a family of algebraic plane
curves depending on the parameters $t,s$, we provide an algorithm for studying
the topology types arising in the family. For this purpose, the algorithm
computes a finite partition of the parameter space so that the topology type of
the family stays invariant over each element of the partition. The ideas
contained in the paper can be seen as a generalization of the ideas in
\cite{JGRS}, where the problem is solved for families of algebraic curves
depending on one parameter, to the two-parameters case.
