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

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. J. Math.**33**(1990), 247-252. MR**1060381 (91h:31011)****[2]**E. M. Andreev,*On convex polyhedra of finite volume in Lobacevskii space*, Mat. Sb. (N.S.)**83**(1970), 256-260; English transl. in Math. U.S.S.R.-Sb.**12**(1979), 255-259. 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), 569-576. MR**752985 (85k:52009)****[5]**A. Beardon and K. Stephenson,*The finite Schwarz-Pick lemma*, preprint.**[6]**L. Bers,*On a theorem of Mori and the definition of quasiconformality*, Trans. Amer. Math. Soc.**84**(1957), 78-84. 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), 23-30. MR**946096 (90e:30008)****[10]**Zheng-Xu 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), 293-295. MR**0422606 (54:10592)****[13]**O. Lehto and K. I. Virtanen,*Quasiconformal mappings in the plane*, Springer-Verlag, 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), 349-360. MR**906396 (90c:30007)****[16]**B. Rodin,*Schwarz's lemma for circle packings*, Invent. Math.**89**(1987), 271-289. 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*, North-Holland, 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 Lichtenstein-Gershgorin integral equation in conformal mapping*. I.*Theory*, Nat. Bur. Standards Appl. Math. Ser.**42**(1955), 7-30. MR**0074121 (17:540a)**

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