  We provide superposition formulae for the six cases of B\"acklund
transformations corresponding to space-like and time-like surfaces in the
3-dimensional pseudo-euclidean space. In each case, the surfaces have constant
negative or positive Gaussian curvature and they correspond to solutions of one
of the following equations: the sine-Gordon, the sinh-Gordon, the elliptic
sine-Gordon and the elliptic sinh-Gordon equation. The superposition formulae
provide infinitely many solutions algebraically, after the first integration of
the B\"acklund transformation. Such transformations and the corresponding
superposition formulae provide solutions of the same hyperbolic equation, while
they show an unusual property for the elliptic equations. The B\"acklund
transformation alternates solutions of the elliptic sinh-Gordon equation with
those of the elliptic sine-Gordon equation and the superposition formulae
provide solutions of the same elliptic equation. Explicit examples and
illustrations are given and their singularities are briefly discussed.

Analysis of PDEs;Differential Geometry