5
$\begingroup$

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?

$\endgroup$
  • 5
    $\begingroup$ as a British English speaker I'm pretty sure 'parameterise' is fine, but thanks for the edit anyway :) $\endgroup$ – Tim Apr 12 '13 at 15:34
4
$\begingroup$

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

$\endgroup$
  • $\begingroup$ cannot believe I didn't think to check Wikipedia, sorry! but thank you very much $\endgroup$ – Tim Apr 11 '13 at 23:21
  • 1
    $\begingroup$ @Tim no problem. $\endgroup$ – Coffee_Table Apr 11 '13 at 23:29
2
$\begingroup$

A circle of radius $a$ centered at $(b,0)$ in the plane $xz$ has the parametric equation

$$x=a\cos(\theta)+b,z=a\sin(\theta),$$ with $\theta$ in the range $[0,2\pi]$ for a full circle.

Now you rotate the plane $xz$ around $z$ by $x\leftarrow x\cos(\phi),y\leftarrow x\sin(\phi)$, with $\phi$ in the range $[0,2\pi]$ for a full turn,

$$x=(a\cos(\theta)+b)\cos(\phi),\\ y=(a\cos(\theta)+b)\sin(\phi),\\ z=a\sin(\theta).$$

If you freeze $\theta$, you get a circle in a plane parallel to $xy$, of the form:

$$x=r\cos(\phi),y=r\sin(\phi).$$

$\endgroup$
2
$\begingroup$

You can plot this surface in WolframAlpha. The correct syntax to use is:

ParametricPlot3D[{(2 Cos[u]+3) Cos[v], (2 Cos[u]+3) Sin[v], 2 Sin[u]}, {u, 0, 2 Pi}, {v, 0, 2 Pi}]

where I have chosen $a = 2$, $b = 3$. The output looks like this.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.