Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

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
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].


References [Enhancements On Off] (What's this?)

  • [1] Dov Aharonov, The hexagonal packing lemma and discrete potential theory, Canad. Math. Bull. 33 (1990), no. 2, 247–252. MR 1060381, 10.4153/CMB-1990-039-5
  • [2] E. M. Andreev, Convex polyhedra of finite volume in Lobačevskiĭ space, Mat. Sb. (N.S.) 83 (125) (1970), 256–260 (Russian). MR 0273510
  • [3] Ivo Babuška, Milan Práger, and Emil Vitásek, Numerical processes in differential equations, In cooperation with R. Radok. Translated from the Czech by Milada Boruvková, Státní Nakladatelství Technické Literatury, Prague; Interscience Publishers John Wiley & Sons, London-New York-Sydney, 1966. MR 0223101
  • [4] 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, 10.4153/CJM-1984-035-1
  • [5] A. Beardon and K. Stephenson, The finite Schwarz-Pick lemma, preprint.
  • [6] Lipman Bers, On a theorem of Mori and the definition of quasiconformality, Trans. Amer. Math. Soc. 84 (1957), 78–84. MR 0083025, 10.1090/S0002-9947-1957-0083025-7
  • [7] I. Carter, Circle packing and conformal mapping, Ph.D. dissertation, Univ. of California at San Diego, 1989.
  • [8] P. Doyle, Oral communication.
  • [9] Lowell J. Hansen, On the Rodin and Sullivan ring lemma, Complex Variables Theory Appl. 10 (1988), no. 1, 23–30. MR 946096
  • [10] Zheng-Xu He, An estimate for hexagonal circle packings, J. Differential Geom. 33 (1991), no. 2, 395–412. MR 1094463
  • [11] 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
  • [12] Fritz John, A criterion for univalency brought up to date, Comm. Pure Appl. Math. 29 (1976), no. 3, 293–295. MR 0422606
  • [13] 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
  • [14] A. Marden and B. Rodin, On Thurston's formulation and proof of Andreev's theorem, preprint.
  • [15] 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
  • [16] Burt Rodin, Schwarz’s lemma for circle packings, Invent. Math. 89 (1987), no. 2, 271–289. MR 894380, 10.1007/BF01389079
  • [17] -, Schwarz's lemma for circle packings. II, J. Differential Geom. 29 (1989).
  • [18] Joe F. Thompson (ed.), Numerical grid generation, North-Holland Publishing Co., New York-Amsterdam, 1982. MR 676679
  • [19] W. P. Thurston, The geometry and topology of $ 3$-manifolds, Princeton Univ. Notes, Princeton, N.J., 1980.
  • [20] -, The finite Riemann mapping theorem, Invited Address, Internat. Sympos. in Celebration of the Proof of the Bierberbach Conjecture, Purdue University, March 1985.
  • [21] 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

Similar Articles

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

DOI: https://doi.org/10.1090/S0002-9947-1992-1081937-X
Keywords: Discrete conformal geometry, circle packing, numerical conformal mapping, grid generation
Article copyright: © Copyright 1992 American Mathematical Society