Find the normal vector or an ellipsoid given the scale vector?

Question

Each point of a unit sphere is conveniently also its own normal. I am starting with a unit sphere and then multiplying each point by a scale vector to create an ellipsoid. I would like to know the normal vector at each vertex. Is there some simple math to get the normal vector given the unit sphere vector and the scale vector?

Answer 1

Yes. There is.

The equation of the ellipsoid after scaling the unit sphere by $a,b,c$ is

$ \dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} + \dfrac{z^2}{c^2} = 1 $

The gradient vector which is also the normal to the surface of the ellipsoid is given by

$ \vec{N} = 2 \left( \dfrac{x}{a^2}, \dfrac{y}{b^2}, \dfrac{z}{c^2} \right) $