  Conformal transformations can be used to obtain the order parameter for
two-dimensional systems at criticality in finite geometries with fixed boundary
conditions on a connected boundary. To the known examples of this class (such
as the disk and the infinite strip) we contribute the case of a rectangle. We
show that the order parameter profile for simply connected boundaries can be
represented as a universal function (independent of the criticality model)
raised to the power eta/2. The universal function can be determined from the
Gaussian model or equivalently a problem in two-dimensional electrostatics. We
show that fitting the order parameter profile to the theoretical form gives an
accurate route to the determination of eta. We perform numerical simulations
for the Ising model and percolation for comparison with these analytic
predictions, and apply this approach to the study of the planar rotor model.
