The convergence of circle packings to the Riemann mapping (original) (raw)
Related papers
Circle packings in the approximation of conformal mappings
Bulletin of the American Mathematical Society, 1990
Connections between circle packings and analytic functions were first suggested by William Thurston [T2], who conjectured that the conformai mapping of a simply connected plane domain Q to the unit disc A could be approximated by manipulating hexagonal circle configurations lying in Q. The conjecture was confirmed by Rodin and Sullivan [RS]. Their proof relies heavily on the hexagonal combinatorics of the circle configurations, a restriction not suggested by the underlying intuition. The purpose of this note is to announce that Thurston's conjecture is true with much weaker combinatoric hypotheses and to outline the proof. The main lines of argument are those developed by Rodin and Sullivan, but the proof is independent. The deepest part of their work-a uniqueness result of Sullivan's which depends on Mostow rigidity-is replaced here by probabilistic arguments. We work in the setting of hyperbolic geometry and make use of the discrete Schwarz-Pick lemma proven in [BS] to understand the behavior of circle configurations. We analyze how curvature distributes itself around a packing as successive differential changes are made to boundary circles, ultimately modelling this process as a random walk. The proof that a certain limiting random walk is recurrent replaces the uniqueness result of Sullivan. Details will appear elsewhere. The author gratefully acknowledges support of the National Science Foundation and the Tennessee Science Alliance. STATEMENT OF THE MAIN RESULT Let P denote a finite collection of circles in the plane having mutually disjoint interiors. Connect centers of tangent circles with euclidean line segments. If a triangulation of a simply connected closed region of the plane results, we say that P is a circle packing.
Circle packings of maps —The Euclidean case
Rendiconti del Seminario Matematico e Fisico di Milano, 1997
In an earlier work, the author extended the Andreev-Koebe-Thurston circle packing theorem. Additionally, a polynomial time algorithm for constructing primal-dual circle packings of arbitrary (essentially) 3-connected maps was found. In this note, additional details concerning surfaces of constant curvature 0 (with special emphasis on planar graphs where a slightly different treatment is necessary) are presented.
An Inverse Problem for Circle Packing and Conformal Mapping
Transactions of the American Mathematical Society, 1992
Figure 1.1 illustrates the fact that if a region is almost packed with circles of radius e in the hexagonal pattern and if the unit disk is packed in an isomorphic pattern with circles of varying radii then, after suitable normalization, the correspondence of circles converges to the Riemann mapping function as e-► 0 (see [15]). In the present paper an inverse of this result is obtained as illustrated by Figure 1.2; namely, if the unit disk is almost packed with ecircles there is an isomorphic circle packing almost filling the region such that, after suitable normalization, the circle correspondence converges to the conformai map of the disk onto the region as e-> 0. Note that this set up yields an approximate triangulation of the region by joining the centers of triples of mutually tangent circles. Since this triangulation is intimately related to the Riemann mapping it may be useful for grid generation [18].
International Journal of Mathematics and Mathematical Sciences, 2005
Given a bounded sequence of integers {d 0 ,d 1 ,d 2 ,...}, 6 ≤ d n ≤ M, there is an associated abstract triangulation created by building up layers of vertices so that vertices on the nth layer have degree d n . This triangulation can be realized via a circle packing which fills either the Euclidean or the hyperbolic plane. We give necessary and sufficient conditions to determine the type of the packing given the defining sequence {d n }.
Directed packings of circles in the plane
Proceedings of the 2nd Croatian Combinatorial Days, 2019
We consider sequential packings of families of circles in the plane whose curvatures are given as members of a sequence of non-negative real numbers. Each such packing gives rise to a sequence of circle centers that might diverge to infinity or remain bounded. We examine the behavior of the sequence of circle centers as a function of the growth rate of the sequence of curvatures. In several special cases we obtain explicit formulas for the coordinates of the limit, while in other cases we obtain accurate estimates.
Circle Packings of Maps in Polynomial Time
European Journal of Combinatorics, 1997
The Andreev-Koebe-Thurston circle packing theorem is generalized and improved in two ways. First, we get 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). Secondly, an algorithm is obtained that given a map M and a rational number ε > 0 finds an ε-approximation for the radii and the coordinates of the centres for the primal-dual circle packing representation of M. The algorithm is polynomial in |E(M)| and log(1/ε). 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 π 2 .