Conformal Maps


Power Transform

\(w=z^2\)

Continuous conformal transform with polar coordinate grid. Radial lines spread uniformily, circle radii scale parabolically (\(r>1\) – expand, \(r=1\) – constant, \(r<1\) – shrink):



Same transform with a Cartesian grid. All coordinate lines bend into parabolae:


Exponential Transform

\(w=e^z\)

Transfrom through non-conformal intermediate states. Horizontal coordinate change into radial (\(log\)), vertical coordinate change into polar


Zhukovsky Transform

\(w=\frac{1}{2}\left(z+z^{-1}\right)\)

Continuous mapping through the Karman-Trefftz generalization. Radial lines bend into hyperbolae, concentric circles transform into ellipses:


Same transform applied to a circle, bending it into an airfoil profile:


Airflow around the wing obtained by transforming solution around a cylinder:

    References

  1. Zhukovsky flow (interactive animation on Shadertoy)

References