  We discuss some aspects of the theory of recognition of two-dimensional
shapes by means of fingerprints of Jordan curves. An interesting approach to
problems on shape recognition suggested by P.~Ebenfelt, D.~Khavinson, and
H.~Shapiro and extended further by M.~Younsi reveals the fact that the
fingerprints of polynomial lemniscates and, more generally, fingerprints of
rational lemniscates can be obtained as solutions to certain functional
equations involving Blaschke products.
  Our main goal here is to develop an approach which relates fingerprints of
Jordan curves composed of arcs of trajectories and orthogonal trajectories of
certain quadratic differentials with solutions of functional equations
involving pullbacks of these quadratic differentials under appropriate Riemann
mapping functions. In particular, we show that the previous results of
P.~Ebenfelt, D.~Khavinson, and H.~Shapiro and the recent results of M.~Younsi
follow from our more general theorems as special cases.
