How to parametrise a surface

Question

How would one parametrise the surface:$$x^2+2y^2+z^2=1$$

I was thinking that you would need to use spherical coordinates, $x=r \sin\theta \cos\phi$, $y=r \sin\theta \sin\phi$ and $z=r\cos\theta$, although, I'm not sure how this works in regards to this specific surface because of the coefficient of the $y$-component.

Any help would be appreciated.

Thanks!

So, $x=\sin\theta\cos\phi,\sqrt2y=\sin\theta\sin\phi, z=\cos\theta $ — lab bhattacharjee, May 10 '17 at 12:40
Hint: Use the transformation $w=\sqrt{2}y$ so that $w^2=2y^2$, parametrize the sphere, and then transform it back. — Michael Burr, May 10 '17 at 12:42
It can give you ideas about other parametrizations (http://mathworld.wolfram.com/Ellipsoid.html) (https://math.stackexchange.com/q/205915) — Jean Marie, May 11 '17 at 09:23
I understand how the parametrisation works now, I think it was more the coefficient of 2 out of the front of the $y$ term that was confusing me. — Nathan Lowe, May 11 '17 at 09:38

score 0 · Answer 1 · answered May 15 '17 at 16:50

To parametrise this surface you could simply write this as : $\left ( x, y, \sqrt{1-x^2-2y^2} \right)$

This is the easiest way ! And from here you can do your Gradient or Jacobian easely. Of course there is some better way to parametrise, but you should plot and see if you can project on the (x,y) plane or something.

How to parametrise a surface

1 Answers1