Circle packings and hyperbolic surfaces of finite type

Abstract

This paper constructs hyperbolic polyhedral metrics via circle packings. We introduce the curvature of circles as a parameter to include all three types of constant curvature curves in the hyperbolic geometry. This provides a unified approach to producing polyhedral metrics for surfaces of broader topological types. The combinatorial total geodesic curvature serves as an effective tool for establishing the existence and uniqueness of the packing.