Abstract
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 correspondencef ɛ 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 off ɛ ; namely, |f′| is the limit of the ratio of the radii of a target circle off ɛ to its source circle. In the present paper we show how to approximatef′ andf″ in terms off ɛ . 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.
Article PDF
Similar content being viewed by others
References
[Ah] D. Aharanov, The hexagonal packing lemma and discrete potential theory,Canad. Math. Bull. 33 (1990), 247–252.
[H] Z.-X. He, An estimate for hexagonal circle packings,J. Differential Geom.,33 (1991), 395–412.
[HR] Z.-X. He and B. Rodin, Convergence of circle packings of finite valence to Riemann mappings, Preprint.
[MR] A. Marden and B. Rodin,On Thurston's Formulation and Proof of Andreev's Theorem, Lecture Notes in Mathematics, Vol. 1435, Springer-Verlag, Berlin, (1990), pp. 103–115.
[R1] B. Rodin, Schwarz's lemma for circle packings, I,Invent. Math.,89 (1987), 271–289.
[R2] B. Rodin, Schwarz's lemma for circle packings, II,J. Differential Geom.,30 (1989), 539–554.
[RS] B. Rodin and D. Sullivan, The convergence of circle packings to the Riemann mapping,J. Differential Geom. 26 (1987), 349–360.
[S1] O. Schramm, Rigidity of infinite (circle) packings,J. Amer. Math. Soc.,4 (1991), 127–149.
[S2] O. Schramm, Existence and uniqueness of packings with specified combinatorics,Israel J. Math. 73 (1991), 321–341.
[T] W. P. Thurston, The finite Riemann mapping theorem, invited address, International Symposium in Celebration of the Proof of the Bieberbach Conjecture, Purdue University, March 1985.
[W] S.E. Warschawski, On the degree of variation in conformal mapping of variable regions,Trans. Amer. Math. Soc. 69 (1950), 335–356.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Doyle, P., He, ZX. & Rodin, B. Second derivatives of circle packings and conformal mappings. Discrete Comput Geom 11, 35–49 (1994). https://doi.org/10.1007/BF02573993
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/BF02573993