Discrete conformal structures on surfaces with boundary (I) -- Classification

In this paper, we introduce the discrete conformal structures on surfaces with boundary in an axiomatic approach, which ensures that the Poincar\'{e} dual of an ideally triangulated surface with boundary has a good geometric structure.Then we classify the discrete conformal structures on surfaces with boundary, which turns out to unify and generalize Guo-Luo's generalized circle packings \cite{GL2}, Guo's vertex scalings \cite{Guo} and Xu's partial discrete conformal structures \cite{Xu22} on surfaces with boundary.This generalizes the results of Glickenstein-Thomas \cite{GT} on closed surfaces to surfaces with boundary. Motivated by \cite{BPS, Zhang-Guo-Zeng-Luo-Yau-Gu}, we further study the relationships between the discrete conformal structures on surfaces with boundary and the hyperbolic trigonometry. Unexpectedly, we find that some subclasses of the discrete conformal structures on surfaces with boundary are closely related to the twisted generalized hyperbolic triangles introduced by Roger-Yang \cite{Roger-Yang}, which does not appear in the case of closed surfaces. Finally, we study the relationships between the discrete conformal structures on surfaces with boundary and the 3-dimensional hyperbolic geometry by constructing ten types of generalized hyperbolic tetrahedra. Some new generalized hyperbolic tetrahedra recently introduced by Belletti-Yang \cite{B-Y} naturally appears in the constructions.