Rodin and Sullivan （1987） 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 ∞.
In this paper, we present a new proof of the uniqueness of Koebe-Andreev-Thurston theorem. Our method is based on the argument principle in complex analysis and reviews the connection between the circle packing theorem and complex analysis.
