  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.
