The Joukowksy Transform is an elegant and simple way to create an airfoil shape.

Let \(C\) be a circle in the complex plane that passes through the point \(z=1\) and encompasses the point \(z=-1\). The transform is defined as

\[ \zeta=z+\frac{1}{z}. \]

We can explore the transform by looking at the circles centred at \((-r,0)\) with \(r<0\) and with radius \(1+r\):

\[ \|z-r\|=1+r \]

or in cartesian coordinates with parameter \(t\):

\begin{align*}
x &= -r+(1+r)\cos(t)\\

y &= (1+r)\sin(t)
\end{align*}

so that \[ (x,y)\rightarrow \left(x+\frac{x}{x^2+y^2},y-\frac{y}{x^2+y^2}\right). \]

To see this in action, move the point \(P\) in this diagram about, ensuring that the point \((-1,0)\) always remains within the circle: