Planar Conformal Mappings of Piecewise Flat Surfaces

Abstract

There is a rich literature in the theory of circle packings on geometric surfaces that from the beginning has exposed intimate connections to the approximation of conformal mappings. Indeed, one of the first publications in the subject, Rodin and Sullivan’s 1987 paper [10], provides a proof of the convergence of a circle packing scheme proposed by Bill Thurston for approximating the Riemann mapping of an arbitrary proper simply-connected domain in ℂ to the unit disk. Bowers and Stephenson’s work in [4], which explains how to apply the Thurston scheme on nonplanar surfaces, may be viewed as a far reaching generalization of his scheme to the setting of arbitrary equilateral surfaces. Further, in [4] Bowers and Stephenson propose a method for uniformizing more general piecewise flat surfaces that necessitates a truly new ingredient, namely, that of inversive distance packings. This inversive distance scheme was introduced in a very preliminary way in [4] with some comments on the difficulty involved in proving that it produces convergence to a conformal map. Even with these difficulties, the scheme has been encoded in Stephenson’s packing software CirclePack and, though all the theoretical ingredients for proving convergence are not in place, it seems to work well in practice. This paper may be viewed as a commentary on and expansion of the discussion of [4]. Our purposes are threefold. First, we carefully describe the inversive distance scheme, which is given only cursory explanation in [4]; second, we give a careful analysis of the theoretical difficulties that require resolution before conformal convergence can be proved; third, we give a gallery of examples illustrating the power of the scheme. We should note here that there are special cases (e.g., tangency or overlapping packings) where the convergence is verified, and our discussion will give a proof of convergence in those cases.