Circle Packing in Euclidean and Hyperbolic Geometries

Given a graph that defines a triangulation of a simply connected surface, it is possible to associate a radius with each vertex so that the vertices represent centers of circles, and the edges denote patterns of tangency. Such a configuration of circles is called a circle packing. We shall give evidence for the existence and uniqueness of circle packings generated by such graphs, as well as an explanation of the algorithms used to find and output a circle packing on the complex plane and hyperbolic disc.