This paper presents a numerical method able to compute all possible configurations of a planar linkage. The procedure is applicable to rigid linkages (i.e., those that can only adopt a finite number of isolated configurations) and to mobile ones (i.e., those that have internal degrees of freedom). The method is based on the fact that this analysis always reduces to finding the roots of a polynomial system of linear, quadratic, and hyperbolic equations, which is here tackled with a new strategy exploiting its structure. The method is conceptually simple, geometric in nature, and easy to implement, yet it provides solutions of the desired accuracy in short computation times. Experiments are included which show its performance on the double butterfly linkage, for which an accurate an complete discretization of its configuration space is obtained