A POLYNOMIAL-TIME CIRCLE PACKING ALGORITHM

The Andreev-Koebe-Thurston circle packing theorem is generalized and improved in two ways. Simultaneous circle packing representations of the map and its dual map are obtained such that any two edges dual to each other cross at the right angle. The necessary and sufficient condition for a map to have such a primal-dual circle packing representation is that its universal cover graph is 3-connected. A polynomial time algorithm is obtained that given such a map M and a rational number epsilon>0 finds an epsilon-approximation for the primal-dual circle packing representation of M. In particular, there is a polynomial time algorithm that produces simultaneous geodesic line convex drawings of a given map and its dual in a surface with constant curvature, so that only edges dual to each other cross.