  We investigate how observations of strong lensing can be used to infer
cosmological parameters, in particular the equation of state of dark energy. We
focus on the growth of the critical lines of lensing clusters with the source
redshift as this behaviour depends on the distance-redshift relation and is
therefore cosmologically sensitive. Purely analytical approaches are generally
insufficient because they rely on axisymmetric mass distributions and thus
cannot take irregular critical curves into account. We devise a numerical
method based on the Metropolis-Hastings algorithm: an elliptical generalization
of the NFW density profile is used to fit a lens model to an observed
configuration of giant luminous arcs while simultaneously optimizing the
geometry. A semi-analytic method, which derives geometric parameters from
critical points, is discussed as a faster alternative. We test the approaches
on mock observations of gravitational lensing by a numerically simulated
cluster. We find that no constraints can be derived from observations of
individual clusters if no knowledge of the underlying mass distribution is
assumed. Uncertainties are improved if a fixed lens model is used for a purely
geometrical optimization, but the choice of a parametric model may produce
strong biases.
