  We provide an algorithm that computes a set of generators for any complete
ideal in a smooth complex surface. More interestingly, these generators admit a
presentation as monomials in a set of maximal contact elements associated to
the minimal log-resolution of the ideal. Furthermore, the monomial expression
given by our method is an equisingularity invariant of the ideal. As an
outcome, we provide a geometric method to compute the integral closure of a
planar ideal and we apply our algorithm to some families of complete ideals.
