CMC-1 surfaces via osculating M\"{o}bius transformations between circle patterns

Given two circle patterns of the same combinatorics in the plane, the M\"{o}bius transformations mapping circumdisks of one to the other induces a $PSL(2,\mathbb{C})$-valued function on the dual graph. Such a function plays the role of an osculating M\"{o}bius transformation and induces a realization of the dual graph in hyperbolic space. We characterize the realizations and obtain a one-to-one correspondence in the cases that the two circle patterns share the same shear coordinates or the same intersection angles. These correspondences are analogous to the Weierstrass representation for surfaces with constant mean curvature $H\equiv 1$ in hyperbolic space. We further establish convergence on triangular lattices.