Incompressibility of domain-filling circle packings (original) (raw)

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.

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].

Computing conformal maps onto circular domains

We show that, given a non-degenerate, finitely connected domain DDD, its boundary, and the number of its boundary components, it is possible to compute a conformal mapping of DDD onto a circular domain \emph{without} prior knowledge of the circular domain. We do so by computing a suitable bound on the error in the Koebe construction (but, again, without knowing the circular domain in advance).

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.

Approximating the Conformal Maps of Elongated Quadrilaterals by Domain Decomposition

Constructive Approximation, 2001

Let Q := f ; z 1 ; z 2 ; z 3 ; z 4 g be a quadrilateral consisting of a Jordan domain and four points z 1 , z 2 , z 3 , z 4 , in counterclockwise order on @ and let m(Q) be the conformal module of Q. Then Q is conformally equivalent to the rectangular quadrilateral fR m(Q) ; 0; 1; 1 + im(Q); im(Q)g; where R m(Q) := f( ; ) : 0 < < 1; 0 < < m(Q)g; in the sense that there exists a unique conformal map f : ! R m(Q) that takes the four points z 1 ; z 2 ; z 3 ; z 4 , respectively onto the four vertices 0, 1, 1 + im(Q), im(Q) of R m(Q) . In this paper we consider the use of a domain decomposition method (DDM) for computing approximations to the conformal map f, in cases where the quadrilateral Q is \long". The method has been studied already but, mainly, in connection with the computation of m(Q). Here we consider certain recent results of Laugesen 12], for the DDM approximation of the conformal map f : ! R m(Q) associated with a special class of quadrilaterals (viz. quadrilaterals whose two non-adjacent boundary segments (z 2 ; z 3 ) and (z 4 ; z 1 ) are parallel straight lines) and seek to extend these results to more general quadrilaterals. By making use of the available DDM theory for conformal modules, we show that the corresponding theory for f can, indeed, be extended to a much wider class of quadrilaterals than those considered by Laugesen. AMS classi cation: 30C30, 65E05.

Circular arc polygons, numerical conformal mappings, and moduli of quadrilaterals

ArXiv, 2021

We study numerical conformal mappings of planar Jordan domains with boundaries consisting of finitely many circular arcs and compute the moduli of quadrilaterals for these domains. Experimental error estimates are provided and, when possible, comparison to exact values or other methods are given. The main ingredients of the computation are boundary integral equations combined with the fast multipole method.

Second derivatives of circle packings and conformal mappings

Discrete & Computational Geometry, 1994

William Thurston conjectured that the Riemann mapping function f from a simply connected region f~ onto the unit disk D can be approximated as follows. Almost fill f~ with circles of radius e packed in the regular hexagonal pattern. There is a combinatorially isomorphic packing of circles in D. The correspondence f~ of e-circles in ~q with circles of varying radii in D should converge to f after suitable normalization. This was proved in [RS], and in [H] an estimate was obtained which led to an approximation of If'L in terms off~; namely, If'l is the limit of the ratio of the radii of a target circle of f, to its source circle. In the present paper we show how to approximate f' and f" in terms of f~. Explicit rates for the convergence to f, f', and f" are obtained. In the special case of convergence to I f'[, the estimate in this paper improves the previously known estimate.