I am trying to understand rotation surfaces.
So the idea I read is that you have some curve $C(v):(x=f(v),z=g(v))$ in the $xz$ plane and you suppose that $x=f>0$ so that the curve does not meet the $z$ axis.
Then, the idea should be that you let the whole space $\mathbb{R}^3$ revolve around the $z$-axis, so that the curve moves following a circle $(\cos u , \sin u )$ in the $xy$ plane with $u \in [0 ,2 \pi)$
So you get a family of curves $C_u: v \mapsto C_u(v)$ parametrised by $u$ which will be the meridians, and to get the whole surface you just let both $u$ and $v$ vary.
With formulas, this idea will be implemented as $$(u,v) \mapsto ( f(v) \cos u , f(v) \sin u, g(v))$$
Q1
Of course this looks plausible, but I would like an explanation as to why this is the parametrization that actually works.
The main question is the one above. The second question is in the following.
If you want to parametrize the torus, you take your curve to be a circle in the $xz$ plane and you rotate following the other circle in the $xy$ plane to get the torus.
Q
Ok. What if I want to get a sphere this way?
If I look at the parametrisation of a sphere in spherical coordinates it looks as the parametrization of a rotational surface where you rotate the curve $C(v)$ given by $v \mapsto (\sin v , \cos v)$ with $v \in [0, \pi].$
This is like a “half circle with the first and second coordinates exchanged by $(u,v) \mapsto (v,u)$ “, what kind of thing is it?
So I understand that if you take half of a circle in $xz$ plane and let it move following a circle in $xy$ plane you get a sphere; but then why you exchange the usual coordinates of the circle?