Say you have a parallelogram in the complex Z plane, orientation and parameter $\lambda$ can be chosen size of sides has to be arbitrary. Can any such parallelogram be mapped to a parallelogram in the w plane(of any size/orientation) using the inverse Joukowksi map?
The Joukowski map is given by $J(z) = z + \lambda^2/z$ where $\lambda >0$ and $J^{-1}(z)=\frac{1}{2}(z+\sqrt{z^2 - 4\lambda^2})$
So far I don't see how this could be possible but I may be mistaken. If it is indeed not possible how could I go about proving this? Any push in the right direction is appreciated!