Clarification of a proof of $RP^3\cong SO(3)$

Question

I have some questions about the proof of $RP^3\cong SO(3)$ by Peter Franek on MSE:

Each rotation in $\Bbb R^3$ is characterized by an "oriented axis" $v\in S^2$ and an angle $\varphi\in [0,\pi]$ and the only relations are $(v,\pi)=(-v,\pi)$ and $(v,0)=(w,0)$ for each $v,w\in S^2$. If you represent $\Bbb RP^3$ as a 3-ball of diameter $\pi$ with identified antipodal points $v\cdot \pi=-v\cdot \pi$ for each $v\in S^2$, then the map $SO(3)\to \Bbb RP^3$ just maps $(v,\varphi)$ to $[v\cdot \varphi]$. The angle $\varphi\,\,\mathrm{mod}\,2\pi$ depends continuously on the rotation and the axis $v$ depends continuously on the rotation whenever $\varphi\neq 0$.

I don't understand how we regard $[v\cdot \varphi]$ as an element of $\Bbb RP^3$. $v\in S^2$ only has three coordinates but an element in $\Bbb RP^3$ should be like $[x,y,z,w]$ right? If his $[v\cdot \varphi]$ really means $[v, \varphi]$ then $[v,0] \ne [v',0]$. Can some clarify the definition of his map for me?

Source https://math.stackexchange.com/a/1688183/185631

Answer 1

$RP^3$ can be represented in different equivalent ways. Most common are

1. The space of lines in $R^4$ that pass through origin
2. The $3$-dimensional sphere $S^3$ with antipodal points identified
3. The 3-dimensional ball $B^3$ with antipodat points on the boundary sphere identified.

Equivalence of 1. and 2. is probably very intuitive (you represent each line by a pair of antipodal points on the sphere).

Equivalence of 2. and 3. can be seen if you take, in the sphere $S^3$, the upper hemisphere $\{(x_1, x_2, x_3, x_4)\, \,| x_4 \geq 0\}$. This is topologically equivalent to a $3$-ball (when $x_4 > 0$), while on the boundary sphere $(x_4=0)$ you identify antipodal points. Taking the upper hemisphere in definition 2. is good enough because if you have a pair of antipodal points in the 3-sphere, at least on of these points will be in the upper hemisphere.

In the above answer, I was using representation 3. of $RP^3$. I should have stated it more explicitly. By $v \varphi$ I meant a point in the 3-ball ($v$ is on the 2-sphere, $\varphi$ is a scalar $\leq 1$). By $[v\varphi]$ I mean the equivalence class of this points, once you make the identification of antipodal points on the boundary sphere of the $3$-ball.

Admittedly, I'm not going into details -- formal proof would probably go different if you consider just topological spaces, manifolds, of smooth manifold structure. I was mainly trying to deliver the intuition.