Generalised model-independent characterisation of strong gravitational lenses I. Theoretical foundations

We extend our model-independent approach for characterising strong gravitational lenses to its most general form to leading order and use the orientation angles of a set of multiple images with respect to their connection line(s) in addition to the relative distances between the images, their ellipticities, and time-delays. For two symmetric images that straddle the critical curve, the orientation angle additionally allows us to determine the slope of the critical curve and a second (reduced) flexion coefficient at the critical point on the connection line between the images. It also allows us to drop the symmetry assumption that the axis of largest image extension is orthogonal to the critical curve. For three images almost forming a giant arc, the degree of assumed image symmetry is also reduced to the most general case, describing image configurations for which the source need not be placed on the symmetry axis of the two folds that unite at the cusp. For a given set of multiple images, we set limits on the applicability of our approach, show which information can be obtained in cases of merging images, and analyse the accuracy achievable due to the Taylor expansion of the lensing potential for the fold case on a galaxy cluster scale Navarro-Frenk-White-profile, a fold and cusp case on a galaxy cluster scale singular isothermal ellipse, and compare the generalised approach with our previously published one. The position of the critical points is reconstructed with less than 5 '' deviation for multiple images closer to the critical points than 30% of the (effective) Einstein radius. The slope of the critical curve at a fold and its shape in the vicinity of a cusp deviate