Discrete & Computational Geometry

, Volume 11, Issue 1, pp 35–49

Second derivatives of circle packings and conformal mappings


  • Peter Doyle
    • Princeton University
  • Zheng-Xu He
    • Princeton University
  • Burt Rodin
    • Department of MathematicsUniversity of California, San Diego

DOI: 10.1007/BF02573993

Cite this article as:
Doyle, P., He, Z. & Rodin, B. Discrete Comput Geom (1994) 11: 35. doi:10.1007/BF02573993


William Thurston conjectured that the Riemann mapping functionf from a simply connected region Ω onto the unit disk\(\mathbb{D}\) can be approximated as follows. Almost fill Ω with circles of radius ɛ packed in the regular hexagonal pattern. There is a combinatorially isomorphic packing of circles in\(\mathbb{D}\). The correspondencefɛ of ɛ-circles in Ω with circles of varying radii in\(\mathbb{D}\) should converge tof after suitable normalization. This was proved in [RS], and in [H] an estimate was obtained which led to an approximation of |f′| in terms offɛ; namely, |f′| is the limit of the ratio of the radii of a target circle offɛ to its source circle. In the present paper we show how to approximatef′ andf″ in terms offɛ. Explicit rates for the convergence tof, f′, andf″ are obtained. In the special case of convergence to |f′|, the estimate in this paper improves the previously known estimate.

Copyright information

© Springer-Verlag New York Inc. 1994