The Existence And Uniqueness Of Circle Packings And Their Algorithm Via Combinatorial Ricci Flows

A circle packing(or circle pattern) is a confguration of circles with a specifed in-tersection in a constant curvature surface. The theory of circle packing involves theconstruction of polyhedra in hyperbolic3-space, the discrete approximation of analyt-ical functions and the development of a discrete analytical function theory, etc.. Ourmain work in this paper includes two respects: frstly, Andreev-Thurston theorem forcircle patterns is discussed using energy function approach. Given a weighted triangula-tion (T, Θ), using the techniques of constructing energy functions on its interior vertices,it is showed that circle patterns with given boundary radii for (T, Θ) is uniquely deter-mined by its cone vector. The existence and uniqueness theorems of univalent andbranched circle patterns in complex plane are proved based on the inequality of conevector respectively. Secondly, a circle packing algorithm via combinatorial Ricci fowsis investigated. Given a triangulation T, we use the idea of combinatorial Ricci fowsdescribe a algorithm of circle packing P realizing T, which possesses simple iterationprocess, and converges exponentially fast to the raddi of circle packing P. This providesa new and efective method for fnding the radii of circle packing.