  Zamolodchikov's recursion relations are used to analyze the existence and
approximations to the classical conformal block in the case of four parabolic
weights. Strong numerical evidence is found that the saddle point momenta
arising in the classical limit of the DOZZ quantum Liouville theory are simply
related to the geodesic length functions of the hyperbolic geometry on the
4-punctured Riemann sphere. Such relation provides new powerful methods for
both numerical and analytical calculations of these functions. The consistency
conditions for the factorization of the 4-point classical Liouville action in
different channels are numerically verified. The factorization yields efficient
numerical methods to calculate the 4-point classical action and, by the
Polyakov conjecture, the accessory parameters of the Fuchsian uniformization of
the 4-punctured sphere.
