On the classification of discrete conformal structures on surfaces

Glickenstein \cite{Glickenstein} and Glickenstein-Thomas \cite{GT} introduced the discrete conformal structures on surfaces in an axiomatic approach and studied its classification. In this paper, we give a full classification of the discrete conformal structures on surfaces, which completes Glickenstein-Thomas’ classification. As a result, we find some new classes of discrete conformal structures on surfaces, including some of the generalized circle packing metrics introduced by Guo-Luo \cite{GL2}. The relationships between the discrete conformal structures on surfaces and the 3-dimensional hyperbolic geometry are also discussed.