ellipsoid

**yuvalroc** · October 27th 2008, 04:46 AM

What is the external normal of an ellipsoid at a certain point (x,y,z)? How do I compute it?

Thanks,
Yuval

**ThePerfectHacker** · October 27th 2008, 07:27 PM

Originally Posted by yuvalroc

What is the external normal of an ellipsoid at a certain point (x,y,z)? How do I compute it?

Thanks,
Yuval

Let $\frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2}=1$ be your ellipsoid. Define the function $F(x,y,z) = \frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2} - 1$ .
Now use the property that $\nabla F(x_0,y_0,z_0)$ is orthogonal to the ellipsoid $F(x,y,z)=0$ .
Thus, $\left( \frac{2x_0}{a}, \frac{2y_0}{b}, \frac{2z_0}{c} \right)$ is orthogonal to the ellipsoid.
It remains to put a $\pm 1$ in front of it if you want it to be internal or external.
Put the right sign and you have your outward normal vector.

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