= Joukowski circle-to-ellipse mapping
{c}
{title2=$z=\rho e^{i\theta}\mapsto(\rho+\rho^{-1})\cos\theta+i(\rho-\rho^{-1})\sin\theta$}
For $\rho>0$, substitution in the <Joukowski map> gives the title parametrization. If $\rho\ne1$, its image is an <ellipse> with semiaxes $\rho+\rho^{-1}$ and $|\rho-\rho^{-1}|$ and <foci> $\pm2$. Reciprocal radii give the same <ellipse> in opposite orientations. At $\rho=1$ the image degenerates to $[-2,2]$, each interior point covered once by each unit <semicircle>.
Back to article page
