I'm trying to derive the parameterization of a sphere from the general parametric equations for a surface of revolution. In particular, I read on wikipedia, that in general, to parameterize a rotating curve around the x axis, it is possible to use the following set of equations:
$$\left( f(z) \cos \theta, f(z) \sin \theta, z \right)$$
using cylindric coordinates for the section circumferences obtained after the rotation of the curve, $r$ is a function of $z$, thus $r = f(z)$. My question is, how, if it is possible, do I derive the classical parameterization of a sphere ($\left( r \sin \theta \cos \phi, r \sin \theta \sin \phi, r \cos \theta \right)$) from this very same set of equations? Do I have to do some kind of conversion between different coordinate systems?
P.S. Thank you in advance for eventual answers and excuse my poor English, I'm still practising it!
Best Answer
The equation of a sphere is $$x^2+y^2+z^2=r^2$$ so we need $$f(z)^2\cos^2\theta+f(z)^2\sin^2\theta+z^2=r^2$$ or $$f(z)^2+z^2=r^2$$ Therefore $$f(z)=\pm\sqrt{r^2-z^2}$$ Setting $z=r\cos\phi$, we then obtain the parametrization $(r\sin\phi\cos\theta,r\sin\phi\sin\theta,r\cos\phi)$. We can set $z=r\cos\phi$ because we know $z\in[-r,r]$ from the equation of the sphere.
This means that the $\theta$ in the surface revolution parametrization is the $\phi$ in the form you have given the usual parametrization of the sphere.