Huang, Xiaojun; Shen, Liang On the convergence of circle packings to the quasiconformal map. (English) Zbl 1212.30092 Acta Math. Sci., Ser. B, Engl. Ed. 29, No. 5, 1173-1181 (2009). 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 \(\infty\). Cited in 1 Document 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 Keywords:circle packing; quasiconformal map; complex dilation Citations:Zbl 0694.30006 PDFBibTeX XMLCite \textit{X. Huang} and \textit{L. Shen}, Acta Math. Sci., Ser. B, Engl. Ed. 29, No. 5, 1173--1181 (2009; Zbl 1212.30092) Full Text: DOI