Circle Packings of Maps in Polynomial Time

The Andreev?Koebe?Thurston circle packing theorem is generalized and improved in two ways. First, we obtain simultaneous circle packings of the map and its dual map so that, in the corresponding straight-line representations of the map and the dual, any two edges dual to each other are perpendicular. Necessary and sufficient condition for a map to have such a primal?dual circle packing representation in a surface of constant curvature is that its universal cover is 3-connected (the map has no `planar? 2-separations). Second, an algorithm is obtained that given a mapMand a rational number ? < 0 finds an ?-approximation for the radii and the co-ordinates of the centres for the primal?dual circle packing representation ofM. The algorithm is polynomial in |E(M)| and log(1/?). In particular, for a map without planar 2-separations on an arbitrary surface, we have a polynomial time algorithm for simultaneous geodesic convex representations of the map and its dual, so that only edges dual to each other cross, and the angles at the crossings are arbitrarily close to ?/2.