The Joukowsky Transform

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: