This paper investigates the trifocal tensor for an affine trinocular rig and defines an affine trifocal tensor. The question of the degrees of freedom of the tensor entries will be addressed, and a novel algorithm to compute the tensor by a linear technique is presented. It will be shown that 4 point or 8 line correspondences along the three images are enough for a reliable computation of the trifocal tensor in the affine case (uncalibrated, weak perspective camera model), in contrast to the projective case, where at least 7 point or 13 line correspondences are needed for a linear computation (and usually with poor results). An analysis of the error in the approximation of a generic trifocal tensor by the affine trifocal tensor is carried out and preliminary experiments with synthetic and real data show the reliability and robustness of the approximation under a wide range of conditions.