Approximations Of Quasiconformal Mapping And Quasisymmetry By Circle Patterns

The theory of circle patterns is originated from the one of classic circle packingscomprised by disjoint disks which has rapid development in recent years and involves theimportant ideas of complex analysis, di?erential geometry and discrete integrable sys-tems. In this thesis, our main work is as follows: First of all, we use SG circle patternsto discuss the discrete approximation of quasiconformal mappings. The approximat-ing solutions of Beltrami equations on a region are directly constructed by using thetechniques of SG circle patterns and conformal welding. It is proved that these approx-imating solutions converge uniformly on compact subsets of the domain to the exactsolutions; Next, we discuss the discrete approximation of quasisymmetry by SG circlepatterns. For a given quasisymmetry induced from K-quasicircle, we apply SG circlepacking to construct its approximating, and prove these discrete approximating mapsconverge uniformly to the quasisymmetry induced from the K-quasicircle.