  We describe an efficient algorithm to compute a pseudotriangulation of a
finite planar family of pairwise disjoint convex bodies presented by its
chirotope. The design of the algorithm relies on a deepening of the theory of
visibility complexes and on the extension of that theory to the setting of
branched coverings. The problem of computing a pseudotriangulation that
contains a given set of bitangent line segments is also examined.
