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 $g$?