  We begin developing tools to compute off-shell string amplitudes with the
recently proposed hyperbolic string vertices of Costello and Zwiebach.
Exploiting the relation between a boundary value problem for Liouville's
equation and a monodromy problem for a Fuchsian equation, we construct the
local coordinates around the punctures for the generalized hyperbolic
three-string vertex and investigate their various limits. This vertex
corresponds to the general pants diagram with three boundary geodesics of
unequal lengths. We derive the conservation laws associated with such vertex
and perform sample computations. We note the relevance of our construction to
the calculations of the higher-order string vertices using the pants
decomposition of hyperbolic Riemann surfaces.
