Parametrization of the sphere and the torus. Is there a way to find easily the parametrization of the sphere and the tore ? I see on wikipedia that for the sphere it's $(x,y,z)=(\sin \theta\cos \varphi,\sin\theta\sin\varphi,\cos\varphi)$ with $\varphi\in [0,\pi]$ and $\theta\in [0,2\pi]$, but I always forget the formula. Same for the torus.
 A: If $\gamma(t) = \bigl(r(t), 0, z(t)\bigr)$ is a parametric curve in the half-plane $x \geq 0$, $y = 0$, then

 $$(t, \theta) \mapsto \bigl(r(t) \cos\theta, r(t) \sin\theta, z(t)\bigr)$$

parametrizes the surface obtained by revolving the image of $\gamma$ about the $z$-axis.

A: Consider the half circle $\{(\sin \theta,0,\cos\theta)\mid \theta\in [0,2\pi]\}$ and apply the rotation around $Oz$ given by $$\begin{pmatrix}\cos\varphi&-\sin\varphi&0\\\sin\varphi&\cos\varphi&0\\0&0&1\end{pmatrix},$$
with $\varphi\in [0,2\pi]$, what we'll give you $$\begin{pmatrix}\cos\varphi&-\sin\varphi&0\\\sin\varphi&\cos\varphi&0\\0&0&1\end{pmatrix}\begin{pmatrix}\sin\theta\\0\\\cos\theta\end{pmatrix}=(\cos\varphi\sin\theta,\sin\varphi\sin\theta,\cos\theta)$$
with $\varphi\in [0,2\pi]$, $\theta\in [0,\pi]$.
I let you do for the Torus.
A: The one and only way to remember a good parametrization imho  is to draw a figure connecting $ \phi, \theta $ to $ x,y, z $.
For the sphere what are the spherical coordinates? For the torus how is one coordinate is added along radius? 
I purposely avoided giving the parametrization as I believe imagination is right  basis of such 3D geometry, you can help yourself better that way.
EDIT1:
The z-coordinate should be $\cos \theta$ and not $\cos \phi $, an error that would never have been made if individual coordinates are related to $\phi, \theta $ by means of a simple sketch.
