latticeTransform[f_: Function, xrange_: Range, yrange_: Range] :=
  (curve = Graphics[BezierCurve[RandomPoint[
       Rectangle[{xrange[[2]], yrange[[2]]}, {xrange[[3]],yrange[[3]]}], 6]]];
   dom = Show[{ParametricPlot[{x, y}, xrange, yrange,
       AspectRatio -> 1], curve}];
   image = ParametricPlot[{Re[f[z]], Im[f[z]]} /. z -> x + I*y, xrange,
     yrange, PlotRange -> All];
   GraphicsGrid[{{dom, image}}, ImageSize -> Large]);