4
$\begingroup$

Let the equation of an ellipsoid be:

$$2x^2+y^2+2z^2=5$$

And the point on the surface of ellipsoid be:

$$(1,1,1)$$

How to compute the normal to the ellipsoid at the point on the surface of ellipsoid?

I read some article says that I can compute it by gradient, but I am not sure how to do it...

$\endgroup$
3
$\begingroup$

If $c$ is a regular value of a smooth function $f$, then $S=f^{-1}(c)$ is a surface for which $\nabla f(p)$ is normal to $T_pS$. Here $f(x,y,z) = 2x^2+y^2+2z^2$ and $p = (1,1,1)$, so the normal to the ellipsoid at $p$ is $$\nabla f(1,1,1) = (4x,2y,4z)\big|_{x=y=z=1} = (4,2,4).$$

$\endgroup$
2
  • 1
    $\begingroup$ just want to make sure. if the equation is ax^2 + by^2 + cz^2 and p = (d,e,f), the normal to the ellipsoid at p will be (2ad, 2be, 2cf). Is this correct? $\endgroup$ – Mars Lee Sep 15 '16 at 1:41
  • 1
    $\begingroup$ Yup, you got it. $\endgroup$ – Ivo Terek Sep 15 '16 at 1:47

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.