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

Thanks,

Yuval

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- Oct 27th 2008, 04:46 AMyuvalrocellipsoid
What is the external normal of an ellipsoid at a certain point (x,y,z)? How do I compute it?

Thanks,

Yuval - Oct 27th 2008, 07:27 PMThePerfectHacker
Let $\displaystyle \frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2}=1$ be your ellipsoid. Define the function $\displaystyle F(x,y,z) = \frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2} - 1$.

Now use the property that $\displaystyle \nabla F(x_0,y_0,z_0)$ is orthogonal to the ellipsoid $\displaystyle F(x,y,z)=0$.

Thus, $\displaystyle \left( \frac{2x_0}{a}, \frac{2y_0}{b}, \frac{2z_0}{c} \right)$ is orthogonal to the ellipsoid.

It remains to put a $\displaystyle \pm 1$ in front of it if you want it to be internal or external.

Put the right sign and you have your outward normal vector.