Computational Methods and Function Theory

, Volume 3, Issue 1, pp 325–347 | Cite as

Curvature Flow in Conformal Mapping

  • Charles R. Collins
  • Tobin A. Driscoll
  • Kenneth Stephenson
Article

Abstract

We use a simple example to introduce a notion of curvature flow in the conformal mapping of polyhedral surfaces. The inquiry was motivated by experiments with discrete conformal maps in the sense of circle packing. We describe the classical theory behind these flows and demonstrate how to modify the Schwarz-Christoffel method to obtain classical numerical confirmation. We close with some additional examples.

Keywords

Circle packing conformal mapping curvature Schwarz-Christoffel 

2000 MSC

52C26 30C30 53A30 65D15 37E35 53A05 

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References

  1. 1.
    A. F. Beardon, A primer on Riemann surfaces, London Math. Soc. Lecture Note Series, vol. 78, Cambridge Univ. Press, Cambridge, 1984.Google Scholar
  2. 2.
    A. F. Beardon, A “regular” pentagonal tiling of the plane, Conformal Geometry and Dynamics 1 (1997), 58–86.MathSciNetCrossRefGoogle Scholar
  3. 3.
    P. L. Bowers and K. Stephenson, Uniformizing dessins and Bely>l maps via circle packing, Amer. Math. Soc., to appear.Google Scholar
  4. 4.
    B. Chow and F. Luo, Combinatorial Ricci flows on surfaces, Preprint, 2002.Google Scholar
  5. 5.
    Ch. R. Collins and K. Stephenson, A circle packing algorithm, Comput. Geom. 25 (2003), 233–256.MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    T. A. Driscoll, A MATLAB toolbox for Schwarz-Christoffel mapping, 1996.Google Scholar
  7. 7.
    T. A. Driscoll and L. N. Trefethen, Schwarz-Christoffel Mapping, vol. 8, Cambridge Univ. Press, Cambridge, New York, 2002.CrossRefGoogle Scholar
  8. 8.
    T. Dubejko and K. Stephenson, Circle packing: Experiments in discrete analytic function theory, Exp. Math. 4 (1995) no.4, 307–348.MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    P. L. Duren, Univalent Functions, vol. 259, Springer-Verlag, New York, 1983.Google Scholar
  10. 10.
    M. K. Hurdal, P. L. Bowers, K. Stephenson, D. W. L. Sumners, K. Rehm, K. Schaper, and D. A. Rottenberg, Quasi-conformally flat mapping the human cerebellum, in: C. Taylor and A. Colchester (eds.), Medical Image Computing and Computer-Assisted Intervention — MICCAI’99, vol. 1679, Springer, Berlin, 1999, 279–286.CrossRefGoogle Scholar
  11. 11.
    J. W. Cannon, W. J. Floyd, and W. Parry, Finite subdivision rules, Conform. Geom. Dyn. 5 (2001), 153–196.MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    B. Rodin and D. Sullivan, The convergence of circle packings to the Riemann mapping, J. Differ. Geom. 26 (1987), 349–360.MathSciNetMATHGoogle Scholar
  13. 13.
    O. Schramm, Scaling limits of loop-erased random walks and uniform spanning trees, Israel J. Math. 118 (2000), 221–288.MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    K. Stephenson, A probabilistic proof of Thurston’s conjecture on circle packings, Rend. Semin. Mat. Fis. Milano 66 (1996), 201–291.MathSciNetCrossRefGoogle Scholar
  15. 15.
    —, Approximation of conformal structures via circle packing, in: N. Papamichael, St. Ruscheweyh, and E. B. Saff (eds.), Computational Methods and Function Theory 1997, Proceedings of the Third CMFT Conference, vol. 11, World Scientific, 1999, 551–582.Google Scholar
  16. 16.
    —, Circle packing and discrete analytic function theory, in: R. Kuhnau (ed.), Handbook of Complex Analysis, Vol. 1: Geometric Function Theory, Elsevier, 2002.Google Scholar
  17. 17.
    W. Thurston, The finite Riemann mapping theorem, 1985, Invited talk, An International Symposium at Purdue University on the occasion of the proof of the Bieberbach conjecture, March 1985.Google Scholar

Copyright information

© Heldermann  Verlag 2003

Authors and Affiliations

  • Charles R. Collins
    • 1
  • Tobin A. Driscoll
    • 2
  • Kenneth Stephenson
    • 1
  1. 1.MathematicsUniversity of TennesseeKnoxvilleUSA
  2. 2.Mathematical SciencesUniversity of DelawareNewarkUSA

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