Joukowski circle-to-ellipse mapping ($z=\rho e^{i\theta }\mapsto (\rho +{\rho }^{-1})\cos \theta +i(\rho -{\rho }^{-1})\sin \theta$)

For $\rho >0$, substitution in the Joukowski map gives the title parametrization. If $\rho \ne 1$, its image is an ellipse with semiaxes $\rho +{\rho }^{-1}$ and $|\rho -{\rho }^{-1}|$ and foci $\pm 2$. 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.