Definition of the central projection to the unit circle

Question

Since I have some contradictory sources, I decided to ask here to have a proper definition of the central projection to the unit circle $p:\mathbb R^2\backslash \{0\}\rightarrow\mathbb S^1$. Should it be given by normalizing each vector in $\mathbb R^2$, that is, sending $(x,y) \mapsto \left( \dfrac{x}{\sqrt{x^2+y^2}}, \dfrac{y}{\sqrt{x^2+y^2}}\right)$? Or rather the angle function, Arctan, which gives the correct angle of a given point with respect to the $x$-axis, i.e. $$p(x,y)=\begin{cases} \tan^{-1}\left(\frac{y}{x}\right), & \mbox{if } x>0 \\ \frac{\pi}{2}, & \mbox{if } x=0,y>0 \\ -\frac{\pi}{2}, & \mbox{if } x=0,y<0 \\ \tan^{-1}\left(\frac{y}{x}\right)+\pi, & \mbox{if } x<0\end{cases}$$

Answer 1

Well, this depends on how $S^1$ is defined. If $S^1$ is defined as the set $\{(x,y)\in\mathbb{R}^2:x^2+y^2=1\}$ then you want to use your first definition, since $p(x,y)$ needs to be an element of that set. If $S^1$ is instead defined as $\mathbb{R}/2\pi\mathbb{Z}$, then you instead want to use your second definition (where the angles you write stand for their equivalence classes mod $2\pi$).