A stereographic projection related question This might be an easy question, but I haven't been able to up come up with a solution.
The image of the map $$f : \mathbb{R} \to \mathbb{R}^2, a \mapsto (\frac{2a}{a^2+1}, \frac{a^2-1}{a^2+1})$$
is the unit circle take away the north pole. $f$ extends to a function $$g: \mathbb{C} \backslash \{i, -i \} \to \mathbb{C}^2. $$  Can anything be said about the image of $g$?

It is the unit circle. Substituting $\tan t$ for $a$ and simplifying, you get
$$ x=\sin 2t, y=-\cos 2t $$ which is the unit circle.
