Parametrisation of the surface a torus

Question

For a calculus question I need to parameterise the surface of the torus generated by rotating the circle given by $(x-b)^2+z^2=a^2$ around the $z$-axis (with $0<a<b$). I've had a go at this, and have come up with $$\vec{r}(\theta,\phi)=((a\cos\theta+b)\cos\phi, (a\cos\theta+b)\sin\phi, a\sin\theta), ~~ \theta,\phi\in [0,2\pi].$$

I haven't done anything like this for a while now, so I've kind of forgotten how it all goes. So I've tried plotting this on WolframAlpha to see if it's correct, but I can't get it to actually plot a surface defined in this way. I have a copy of Matlab, is there any way I could plot this surface on there to see if it's correct? Basically how can I check if I'm right or not? Failing that, could somebody please just point out if I'm wrong?

Answer 1

Your formula is absolutely correct, just look here for verification.

Answer 2

Here is the MATLAB code:

[phi,theta] = meshgrid(linspace(0,2*pi),linspace(0,2*pi));

r = 1; b = 3;

x = (r.*cos(theta)+b).*cos(phi);
y = (r.*cos(theta)+b).*sin(phi);
z = (r.*sin(theta));
s = surf(x,y,z);

alpha(s,0.5)

shading flat

axis equal

Run this in script (or quickly in Command Window).