Bounded Degree Circle Packings And Conformal Mappings

Circle packings as discrete analogs of conformal mappings is a fast-developingfield of research on border of analysis and geometry. Recent progress was initiated byThurston's idea about the approximation of the Riemann mapping by hexagonal circlepackings in 1985. Shortly, Rodin and Sullivan proved the convergence of Thurston'sscheme, which gives a new discrete geometry view of the Riemann mapping. Afterthen, much research on circle packings and their applications followed. For the studyof circle configuration, classical circle packings comprised by disjoint open disks werelater generalized to circle patterns, where the disks may overlap. In this thesis, ourmain work is as follows. first, we use bounded degree circle packings to study theRiemann mapping, i.e., the convergence of the hexagonal circle packings is generalizedto that of circle packings with bounded degree. It is proved that the first two derivativesof the bounded degree circle packings converge to the Riemann mapping function's firsttwo derivatives, respectively. Moreover, it is given explicit estimates for the rates ofconvergence. Next, we investigate conformal mappings by the method of SG circlepatterns. Given a conformal mapping f from a bounded simply-connected region ontoanother one, its approximating solutions are constructed by the techniques of SG circlepatterns. After a suitable normalization of SG circle patterns, it is proved that theseapproximating solutions converge to f.