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On the convergence of circle packings to the quasiconformal map. (English) Zbl 1212.30092

Summary: B. Rodin and D. Sullivan [J. Differ. Geom. 26, 349–360 (1987; Zbl 0694.30006)] proved Thurston’s conjecture that a scheme based on the circle packing theorem converges to the Riemann mapping, thereby proved a refreshing geometric view of the Riemann mapping theorem. Naturally, we consider to use the ellipses to pack the bounded simply connected domain and obtain similarly a sequence simplicial homeomorphism between the ellipse packing and the circle packing. In this paper, we prove that these simplicial homeomorphism approximate a quasiconformal mapping from the bounded simply connected domain onto the unit disk with the modulus of their complex dilatations tending to 1 almost everywhere in the domain when the ratio of the longer axis and shorter axis of the ellipse tending to \(\infty\).

MSC:

30C85 Capacity and harmonic measure in the complex plane
52C15 Packing and covering in \(2\) dimensions (aspects of discrete geometry)
30C35 General theory of conformal mappings

Citations:

Zbl 0694.30006
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