Cross-Cap Rotation

<< "Graphics`Animation`"f[x_, y_] := ((x y)/((x^2) + (y^2)))x = r Cos[theta]; y = r Sin[theta]; z = Simplify[f[x, y]]; ParametricPlot3D[{x, y, z}, {r, 0, 1}, {theta, 0, 2 π}, 
      BoxRatios -> {1.5, 1.5, 1}, PlotPoints -> {8, 41}, 
      ViewPoint -> {-1.5, -4, 2}, Boxed -> False, 
      Axes -> None, DisplayFunction -> Identity]SpinShow[%, Frames -> 16, SpinRange -> {0, 180 Degree}]