Circle Packing Theorem - Relations With Conformal Mapping Theory

Relations With Conformal Mapping Theory

A conformal map between two open sets in the plane or in a higher dimensional space is a continuous function from one set to the other that preserves the angles between any two curves. The Riemann mapping theorem, formulated by Bernhard Riemann in 1851, states that, for any two open topological disks in the plane, there is a conformal map from one disk to the other. Conformal mappings have applications in mesh generation, map projection, and other areas. However, it is not always easy to construct a conformal mapping between two given domains in an explicit way.

At the Bieberbach conference in 1985, William Thurston conjectured that circle packings could be used to approximate conformal mappings. More precisely, Thurston used circle packings to find a conformal mapping from an arbitrary open disk A to the interior of a circle; the mapping from one topological disk A to another disk B could then be found by composing the map from A to a circle with the inverse of the map from B to a circle.

Thurston's idea was to pack circles of some small radius r in a hexagonal tessellation of the plane, within region A, leaving a narrow region near the boundary of A, of width r, where no more circles of this radius can fit. He then constructs a maximal planar graph G from the intersection graph of the circles, together with one additional vertex adjacent to all the circles on the boundary of the packing. By the circle packing theorem, this planar graph can be represented by a circle packing C in which all the edges (including the ones incident to the boundary vertex) are represented by tangencies of circles. The circles from the packing of A correspond one-for-one with the circles from C, except for the boundary circle of C which corresponds to the boundary of A. This correspondence of circles can be used to construct a continuous function from A to C in which each circle and each gap between three circles is mapped from one packing to the other by a Möbius transformation. Thurston conjectured that, in the limit as the radius r approaches zero, the functions from A to C constructed in this way would approach the conformal function given by the Riemann mapping theorem.

Thurston's conjecture was proven by Rodin & Sullivan (1987). More precisely, they showed that, as n goes to infinity, the function f_n determined using Thurston's method from hexagonal packings of radius-1/n circles converges uniformly on compact subsets of A to a conformal map from A to C.

Despite the success of Thurston's conjecture, practical applications of this method have been hindered by the difficulty of computing circle packings and by its relatively slow convergence rate. However it has some advantages when applied to non-simply-connected domains and in selecting initial approximations for numerical techniques that compute Schwarz–Christoffel mappings, a different technique for conformal mapping of polygonal domains.