An inverse problem for circle packing and conformal mapping
Authors:
Ithiel Carter and Burt Rodin
Journal:
Trans. Amer. Math. Soc. 334 (1992), 861875
MSC:
Primary 52C15; Secondary 30C20, 30C30, 51M15
MathSciNet review:
1081937
Fulltext 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 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 (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 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 . 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].
 [1]
Dov
Aharonov, The hexagonal packing lemma and discrete potential
theory, Canad. Math. Bull. 33 (1990), no. 2,
247–252. MR 1060381
(91h:31011), http://dx.doi.org/10.4153/CMB19900395
 [2]
E.
M. Andreev, Convex polyhedra of finite volume in
Lobačevskiĭ\ space, Mat. Sb. (N.S.) 83
(125) (1970), 256–260 (Russian). MR 0273510
(42 #8388)
 [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, LondonNew YorkSydney, 1966. MR 0223101
(36 #6150)
 [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
(85k:52009), http://dx.doi.org/10.4153/CJM19840351
 [5]
A. Beardon and K. Stephenson, The finite SchwarzPick 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
(18,646d), http://dx.doi.org/10.1090/S00029947195700830257
 [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
(90e:30008)
 [10]
ZhengXu
He, An estimate for hexagonal circle packings, J. Differential
Geom. 33 (1991), no. 2, 395–412. MR 1094463
(92b:52039)
 [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
WileyInterscience Publication. MR 822470
(87h:30002)
 [12]
Fritz
John, A criterion for univalency brought up to date, Comm.
Pure Appl. Math. 29 (1976), no. 3, 293–295. MR 0422606
(54 #10592)
 [13]
O.
Lehto and K.
I. Virtanen, Quasiconformal mappings in the plane, 2nd ed.,
SpringerVerlag, New YorkHeidelberg, 1973. Translated from the German by
K. W. Lucas; Die Grundlehren der mathematischen Wissenschaften, Band 126.
MR
0344463 (49 #9202)
 [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 (90c:30007)
 [16]
Burt
Rodin, Schwarz’s lemma for circle packings, Invent.
Math. 89 (1987), no. 2, 271–289. MR 894380
(88h:11043), http://dx.doi.org/10.1007/BF01389079
 [17]
, Schwarz's lemma for circle packings. II, J. Differential Geom. 29 (1989).
 [18]
Joe
F. Thompson (ed.), Numerical grid generation, NorthHolland
Publishing Co., New YorkAmsterdam, 1982. MR 676679
(83j:65004b)
 [19]
W. P. Thurston, The geometry and topology of 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 LichtensteinGershgorin
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
(17,540a)
 [1]
 Dov Aharonov, The hexagonal packing lemma and discrete potential theory, Canad. J. Math. 33 (1990), 247252. MR 1060381 (91h:31011)
 [2]
 E. M. Andreev, On convex polyhedra of finite volume in Lobacevskii space, Mat. Sb. (N.S.) 83 (1970), 256260; English transl. in Math. U.S.S.R.Sb. 12 (1979), 255259. MR 0273510 (42:8388)
 [3]
 I. Babuska, M. Práger, and E. Vitásek, Numerical processes in differential equations, Wiley, London, 1966, 351 pp. MR 0223101 (36:6150)
 [4]
 I. Bárány, Z. Füredi, and J. Pach, Discrete convex functions and proof of the six circle conjecture of Fejes Tóth, Canad. J. Math. 36 (1984), 569576. MR 752985 (85k:52009)
 [5]
 A. Beardon and K. Stephenson, The finite SchwarzPick lemma, preprint.
 [6]
 L. Bers, On a theorem of Mori and the definition of quasiconformality, Trans. Amer. Math. Soc. 84 (1957), 7884. MR 0083025 (18:646d)
 [7]
 I. Carter, Circle packing and conformal mapping, Ph.D. dissertation, Univ. of California at San Diego, 1989.
 [8]
 P. Doyle, Oral communication.
 [9]
 L. J. Hanson, On the Rodin and Sullivan ring lemma, Complex Variables Theory Appl. 10 (1988), 2330. MR 946096 (90e:30008)
 [10]
 ZhengXu He, An estimate for hexagonal circle packings, J. Differential Geom. (to appear). MR 1094463 (92b:52039)
 [11]
 P. Henrici, Applied and computational complex analysis. Vol. III, Wiley, New York and London, 1986, 637 pp. MR 822470 (87h:30002)
 [12]
 F. John, A criterion for univalency brought up to date, Comm. Pure Appl. Math. 29 (1976), 293295. MR 0422606 (54:10592)
 [13]
 O. Lehto and K. I. Virtanen, Quasiconformal mappings in the plane, SpringerVerlag, New York and Berlin, 1973. MR 0344463 (49:9202)
 [14]
 A. Marden and B. Rodin, On Thurston's formulation and proof of Andreev's theorem, preprint.
 [15]
 B. Rodin and D. Sullivan, The convergence of circle packings to the Riemann mapping, J. Differential Geom. 26 (1987), 349360. MR 906396 (90c:30007)
 [16]
 B. Rodin, Schwarz's lemma for circle packings, Invent. Math. 89 (1987), 271289. MR 894380 (88h:11043)
 [17]
 , Schwarz's lemma for circle packings. II, J. Differential Geom. 29 (1989).
 [18]
 J. F. Thompson (editor), Numerical grid generation, NorthHolland, New York and Amsterdam, 1982. MR 676679 (83j:65004b)
 [19]
 W. P. Thurston, The geometry and topology of 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 LichtensteinGershgorin integral equation in conformal mapping. I. Theory, Nat. Bur. Standards Appl. Math. Ser. 42 (1955), 730. MR 0074121 (17:540a)
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:
http://dx.doi.org/10.1090/S0002994719921081937X
PII:
S 00029947(1992)1081937X
Keywords:
Discrete conformal geometry,
circle packing,
numerical conformal mapping,
grid generation
Article copyright:
© Copyright 1992
American Mathematical Society
