  Minimal area surfaces in AdS$_3$ ending on a given curve at the boundary are
dual to planar Wilson loops in N=4 SYM. In previous work it was shown that the
problem of finding such surfaces can be recast as the one of finding an
appropriate parameterization of the boundary contour that corresponds to
conformal gauge. A. Dekel was able to find such reparameterization in a
perturbative expansion around a circular contour. In this work we show that for
more general contours such reparameterization can be found using a numerical
procedure that does not rely on a perturbative expansion. This provides further
checks and applications of the integrability method. An interesting property of
the method is that it uses as data the Schwarzian derivative of the contour and
therefore it has manifest global conformal invariance. Finally, we apply Shanks
transformation to extend the near circular expansion to larger deformations,
the results are in agreement with the new method.
