  We introduce the Nonlinear Cauchy-Riemann equations as B\"{a}cklund
transformations for several nonlinear and linear partial differential
equations. From these equations we treat in details the Laplace and the
Liouville equations by deriving general solution for the nonlinear Liouville
equation. By M\"{o}bius transformation we relate solutions for the Poincare
model of hyperbolic geometry, the Klein model in half-plane and the
pseudo-sphere. Conformal form of the constant curvature metrics in these
geometries, stereographic projections and special solutions are discussed. Then
we introduce the hyperbolic analog of the Riemann sphere, which we call the
Riemann pseudosphere. We identify point at infinity on this pseudosphere and
show that it can be used in complex analysis as an alternative to usual Riemann
sphere to extend the complex plane. Interpretation of symmetric and antipodal
points on both, the Riemann sphere and the Riemann pseudo-sphere, are given. By
M\"{o}bius transformation and homogenous coordinates, the most general solution
of Liouville equation as discussed by Crowdy is derived.
