Accurate Circle Conngurations and Numerical Conformal Mapping in Polynomial Time

| According to a remarkable re-interpretation of a theorem of E.M. Andreev (1970) by W.P. Thurston (1982), there is a unique (up to inversive transformations) packing of interior-disjoint circles in the plane, whose contact graph is any given polyhedral graph G, and such that an analagous \dual" circle packing simultaneously exists, whose contact graph is the planar dual graph G , and such that the primal and dual circle packin1qDgs have the same set of tangency points and the primal circles are orthogonal to the dual ones at these tangency points. This note shows that relatively and absolutely accurate coordinates for the primal and dual circles may be obtained in time polynomial in N, the number of vertices of the polyhedral graph, and D, the number of decimals of accuracy desired. Consequently one may also accurately \midscribe" a polyhedron { and simultaneously its dual { in polynomial time. Also consequently, one may implement Riemann's conformal mapping theorem numerically, in polynomial time with provable accuracy. Our result is obtained by generalizing and refor-mulating ideas found in the doctoral thesis of Walter Brr agger Math. Institut, Rheinsprung 21, CH-4051 Basel, Feb. 1991] to reduce our problem to maximizing a smooth convex function. This maximization problem is then solved by using Khachian's \ellipsoid method" or Vaidya's algorithm. 1 Background I T WILL BE ASSUMED that the reader is familiar with the basics of graph theory, hyperbolic geometry, and convex programming, and with the notion of \inver-sions" about Euclidean spheres. The uninitiated reader may consult 5, 18, 8, 9, 1], or, should he be lucky enough to obtain a copy, most of the relevant ideas and terminology in these areas are discussed in 13]. The version of the Andreev-Thurston theorem which we are interested in states that 1 Theorem 1 Andreev-Thurston in plane] There is a unique (up to inversive transformations) packing of interior-disjoint circles in the plane, whose contact graph is any given polyhedral graph G. If G is maximal pla-nar, this circle connguration is unique up to inversive transformations of the plane.] Furthermore an analagous \dual" circle packing simultaneously exists, whose contact graph is the planar dual graph G , and such that the primal and dual circles have the same set of tangency points and the primal circles are orthogonal to the dual ones at these tangency points. A graph is \polyhedral" ii it is 3-connected and pla-nar (a famous theorem …