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We provide superposition formulae for the six cases of B\"acklundtransformations corresponding to space-like and time-like surfaces in the3-dimensional pseudo-euclidean space. In each case, the surfaces have constantnegative or positive Gaussian curvature and they correspond to solutions of oneof the following equations: the sine-Gordon, the sinh-Gordon, the ellipticsine-Gordon and the elliptic sinh-Gordon equation. The superposition formulaeprovide infinitely many solutions algebraically, after the first integration ofthe B\"acklund transformation. Such transformations and the correspondingsuperposition formulae provide solutions of the same hyperbolic equation, whilethey show an unusual property for the elliptic equations. The B\"acklundtransformation alternates solutions of the elliptic sinh-Gordon equation withthose of the elliptic sine-Gordon equation and the superposition formulaeprovide solutions of the same elliptic equation. Explicit examples andillustrations are given and their singularities are briefly discussed.

Superposition formulae for the geometric B\"acklund transformations of the hyperbolic and elliptic sine-Gordon and sinh-Gordon equations