An inverse problem for circle packing and conformal mapping

Authors:
Ithiel Carter and Burt Rodin

Journal:
Trans. Amer. Math. Soc. **334** (1992), 861-875

MSC:
Primary 52C15; Secondary 30C20, 30C30, 51M15

DOI:
https://doi.org/10.1090/S0002-9947-1992-1081937-X

MathSciNet review:
1081937

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Figure 1.1 illustrates the fact that if a region is almost packed with circles of radius $\varepsilon$ 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 $\varepsilon \to 0$ (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 $\varepsilon$-circles there is an isomorphic circle packing almost filling the region such that, after suitable normalization, the circle correspondence converges to the conformal map of the disk onto the region as $\varepsilon \to 0$. 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].

- Dov Aharonov,
*The hexagonal packing lemma and discrete potential theory*, Canad. Math. Bull.**33**(1990), no. 2, 247–252. MR**1060381**, DOI https://doi.org/10.4153/CMB-1990-039-5 - E. M. Andreev,
*Convex polyhedra of finite volume in Lobačevskiĭ space*, Mat. Sb. (N.S.)**83 (125)**(1970), 256–260 (Russian). MR**0273510** - Ivo Babuška, Milan Práger, and Emil Vitásek,
*Numerical processes in differential equations*, Státní Nakladatelství Technické Literatury, Prague; Interscience Publishers John Wiley & Sons, London-New York-Sydney, 1966. In cooperation with R. Radok; Translated from the Czech by Milada Boruvková. MR**0223101** - Imre Bárány, Zoltán Füredi, and János Pach,
*Discrete convex functions and proof of the six circle conjecture of Fejes Tóth*, Canad. J. Math.**36**(1984), no. 3, 569–576. MR**752985**, DOI https://doi.org/10.4153/CJM-1984-035-1
A. Beardon and K. Stephenson, - Lipman Bers,
*On a theorem of Mori and the definition of quasiconformality*, Trans. Amer. Math. Soc.**84**(1957), 78–84. MR**83025**, DOI https://doi.org/10.1090/S0002-9947-1957-0083025-7
I. Carter, - Lowell J. Hansen,
*On the Rodin and Sullivan ring lemma*, Complex Variables Theory Appl.**10**(1988), no. 1, 23–30. MR**946096**, DOI https://doi.org/10.1080/17476938808814284 - Zheng-Xu He,
*An estimate for hexagonal circle packings*, J. Differential Geom.**33**(1991), no. 2, 395–412. MR**1094463** - Peter Henrici,
*Applied and computational complex analysis. Vol. 3*, Pure and Applied Mathematics (New York), John Wiley & Sons, Inc., New York, 1986. Discrete Fourier analysis—Cauchy integrals—construction of conformal maps—univalent functions; A Wiley-Interscience Publication. MR**822470** - Fritz John,
*A criterion for univalency brought up to date*, Comm. Pure Appl. Math.**29**(1976), no. 3, 293–295. MR**422606**, DOI https://doi.org/10.1002/cpa.3160290304 - O. Lehto and K. I. Virtanen,
*Quasiconformal mappings in the plane*, 2nd ed., Springer-Verlag, New York-Heidelberg, 1973. Translated from the German by K. W. Lucas; Die Grundlehren der mathematischen Wissenschaften, Band 126. MR**0344463**
A. Marden and B. Rodin, - Burt Rodin and Dennis Sullivan,
*The convergence of circle packings to the Riemann mapping*, J. Differential Geom.**26**(1987), no. 2, 349–360. MR**906396** - Burt Rodin,
*Schwarz’s lemma for circle packings*, Invent. Math.**89**(1987), no. 2, 271–289. MR**894380**, DOI https://doi.org/10.1007/BF01389079
---, - Joe F. Thompson (ed.),
*Numerical grid generation*, Elsevier, New York, 1982. Appl. Math. Comput. 10/11 (1982). MR**675778**
W. P. Thurston, - S. E. Warschawski,
*On the solution of the Lichtenstein-Gershgorin integral equation in conformal mapping. I. Theory*, Experiments in the computation of conformal maps, National Bureau of Standards Applied Mathematics Series, No. 42, U. S. Government Printing Office, Washington, D. C., 1955, pp. 7–29. MR**0074121**

*The finite Schwarz-Pick lemma*, preprint.

*Circle packing and conformal mapping*, Ph.D. dissertation, Univ. of California at San Diego, 1989. P. Doyle, Oral communication.

*On Thurston’s formulation and proof of Andreev’s theorem*, preprint.

*Schwarz’s lemma for circle packings*. II, J. Differential Geom.

**29**(1989).

*The geometry and topology of*$3$-

*manifolds*, Princeton Univ. Notes, Princeton, N.J., 1980. ---,

*The finite Riemann mapping theorem*, Invited Address, Internat. Sympos. in Celebration of the Proof of the Bierberbach Conjecture, Purdue University, March 1985.

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC:
52C15,
30C20,
30C30,
51M15

Retrieve articles in all journals with MSC: 52C15, 30C20, 30C30, 51M15

Additional Information

Keywords:
Discrete conformal geometry,
circle packing,
numerical conformal mapping,
grid generation

Article copyright:
© Copyright 1992
American Mathematical Society