  Recent results using inverse scattering techniques interpret every solution
$\phi (x,y)$ of the sine-Gordon equation as a non-linear superposition of
solutions along the axes $x=0$ and $y=0$. Here we provide a geometric method of
integration, as well as a geometric interpretation. Specifically, every weakly
regular surface of Gauss curvature $K=-1$, in arc length asymptotic line
parametrization, is uniquely determined by the values $\phi(x,0)$ and
$\phi(0,y)$ of its coordinate angle along the axes. Based on a generalized
Weierstrass pair that depends only on these values, we prove that to each such
unconstrained pair of differentiable functions, there corresponds uniquely an
associated family of pseudospherical immersions; we construct these immersions
explicitely.
