This report addresses the problem of computing the parametrization of the intersection of deformable quadratic algebraic surfaces (quadrics) in projective space. It also presents an automatic method for describing the evolution in time of the topology of the intersection. The work is based on the results from \cite{QI,QI2}, which offer an exact parametrization of the intersection of two quadrics with rational coefficients of arbitrary size. This parametrization is rational when one exists, and its coefficients are almost as rational as possible \cite{QI2}. \comment{It is intended to take advantage of the efficiency of the algorithms and of the quasi-optimality of the parametrization.}