Geometriae Dedicata

, Volume 137, Issue 1, pp 163–197 | Cite as

Approximation of conformal mappings by circle patterns

  • Ulrike BückingEmail author
Original Paper


A circle pattern is a configuration of circles in the plane whose combinatorics is given by a planar graph G such that to each vertex of G corresponds a circle. If two vertices are connected by an edge in G, the corresponding circles intersect with an intersection angle in (0, π). Two sequences of circle patterns are employed to approximate a given conformal map g and its first derivative. For the domain of g we use embedded circle patterns where all circles have the same radius decreasing to 0 and with uniformly bounded intersection angles. The image circle pattern has the same combinatorics and intersection angles and is determined from boundary conditions (radii or angles) according to the values of g′ (|g′| or arg g′). For quasicrystallic circle patterns the convergence result is strengthened to C -convergence on compact subsets.


Circle pattern Convergence Conformal mappings Discrete Laplacian 

Mathematics Subject Classification (2000)

30G25 30E10 52C26 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Bobenko A.I., Mercat C., Suris Y.B. (2005) Linear and nonlinear theories of discrete analytic functions. Integrable structure and isomonodromic Green’s function. J. Reine angew. Math. 583: 117–161zbMATHMathSciNetGoogle Scholar
  2. 2.
    Bobenko, A.I., Schröder, P., Sullivan, J.M., Ziegler, G.M. (eds.): Discrete differential geometry. In: Oberwolfach Seminars, vol. 38. Birkhäuser, Basel (2008)Google Scholar
  3. 3.
    Bobenko A.I., Springborn B.A. (2004) Variational principles for circle patterns and Koebe’s theorem. Trans. Am. Math. Soc. 356: 659–689zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Bobenko, A.I., Suris, Y.B.: Discrete differential geometry. The integrable structure (to appear in 2008)Google Scholar
  5. 5.
    Bücking, U.: Approximation of conformal mappings by circle patterns and discrete minimal surfaces. Ph.D. thesis, Technische Universität Berlin (2007). Published online at
  6. 6.
    Carter I., Rodin B. (1992) An inverse problem for circle packing and conformal mapping. Trans. Am. Math. Soc. 334: 861–875zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Duffin R.J. (1953) Discrete potential theory. Duke Math. J. 20: 233–251zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Duffin R.J. (1968) Potential theory on a rhombic lattice. J. Comb. Theory 5: 258–272zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Duneau M., Katz A. (1985) Quasiperiodic patterns. Phys. Rev. Lett. 54: 2688–2691CrossRefMathSciNetGoogle Scholar
  10. 10.
    Gähler F., Rhyner J. (1986) Equivalence of the generalized grid and projection methods for the construction of quasiperiodic tilings. J. Phys. A 19: 267–277zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    He Z.-X. (1999) Rigidity of infinite disk patterns. Ann. Math. 149: 1–33zbMATHCrossRefGoogle Scholar
  12. 12.
    He Z.-X., Schramm O. (1996) On the convergence of circle packings to the Riemann map. Invent. Math. 125: 285–305zbMATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    He Z.-X., Schramm O. (1998) The C -convergence of hexagonal disk packings to the Riemann map. Acta Math. 180: 219–245zbMATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Kenyon R. (2002) The Laplacian and Dirac operators on critical planar graphs. Invent. Math. 150: 409–439zbMATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Lan S.Y., Dai D.Q. (2007) The C -convergence of SG circle patterns to the Riemann mapping. J. Math. Anal. Appl. 332: 1351–1364zbMATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Matthes D. (2005) Convergence in discrete Cauchy problems and applications to circle patterns. Conform. Geom. Dyn. 9: 1–23zbMATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Mercat C. (2001) Discrete Riemann surfaces and the Ising model. Commun. Math. Phys. 218: 177–216zbMATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Rodin B., Sullivan D. (1987) The convergence of circle packings to the Riemann mapping. J. Diff. Geom. 26: 349–360zbMATHMathSciNetGoogle Scholar
  19. 19.
    Saloff-Coste L. (1997) Some inequalities for superharmonic functions on graphs. Potential Anal. 6: 163–181zbMATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    Schramm O. (1997) Circle patterns with the combinatorics of the square grid. Duke Math. J. 86: 347–389zbMATHCrossRefMathSciNetGoogle Scholar
  21. 21.
    Senechal, M.: Quasicrystals and Geometry. Cambridge University Press (1995)Google Scholar
  22. 22.
    Springborn, B.A.: Variational principles for circle patterns. Ph.D. thesis, Technische Universität Berlin (2003). Published online at
  23. 23.
    Stephenson K. (2005) Introduction to circle packing: the theory of discrete analytic functions. Cambridge University Press, New YorkzbMATHGoogle Scholar
  24. 24.
    Thurston, B.: The finite Riemann mapping theorem (1985). Invited address at the International Symposioum in Celebration of the proof of the Bieberbach Conjecture, Purdue UniversityGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2008

Authors and Affiliations

  1. 1.Fakultät II – Institut für MathematikTechnische Universität BerlinBerlinGermany

Personalised recommendations