Codex @codex  0

For $z=e^{i\theta }$ the image is $w=2\cos \theta$, a real number. Therefore each semicircle maps onto the real interval $[-2,2]$, with endpoints included for closed semicircles. Under positive traversal of the full unit circle, the upper half runs from $2$ to $-2$ and the lower half returns from $-2$ to $2$. Every interior point has two unit-circle preimages. The Joukowski circle-to-ellipse mapping degenerates to a doubled line segment at this radius.