I was assigned to find the line element ($ds^2$) of a torus. In order to do so, the first step is to take the partial derivates of the following parametrization:
$x(u, v) = (c + a\cos(v))\cos(u)$
$y(u, v)=(c+acos(v))sin(u)$
$z(u, v)=asin(v)$
What caught my attention is that we only consider the two angular variables ($u$, $v$) and not $a$ and $c$ unlike something like a cylinder ($\rho$,$\theta$,$z$) or a sphere ($r$,$\theta$,$\phi$) where three variables are considered.
So why is the parametrization of a torus a function of two variables and not three($a$, $u$, $v$) ($c$, $u$, $v$) or four ($a$, $c$, $u$, $v$)?
Thank you in advance.
No, for a cylinder, you only use two of your three cylindrical coordinates, namely $(\theta,z)$. For a sphere, you only use two of your three spherical coordinates, namely $(\theta,\phi)$. In both cases, the "radius" coordinate is constant. In general, to call an object a surface, you should have a parametrization (with some properties) by two independent variables.
You can have three, four, or — indeed — arbitrarily many dependent variables. For example, here's a parametrization of a torus in $4$-space: $$(x,y,z,w) = (\cos u,\sin u, \cos v, \sin v).$$