We present in this paper a linear time algorithm for the computation of some distance functions between convex polygons in R² in the general case where they can intersect. The interest for this problem comes from applications in robot vision, pattern recognition and contour fitting, see Cox, Maitre, Minoux and Ribeiro [1989]. The algorithm is based on previous work of Atallah [1983] for the computation of the Hausdorff distance between disjoint convex polygons. Computational results are presented.