TITLE: Parametrization for the ellipsoids
QUESTION [5 upvotes]: Can someone help to describe some possible parametrizations for the ellipsoid:
$$\frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2} = 1?$$
I am thinking polar coordinates, but there may be the concept of steographic projecting (not sure how to apply it here), and not sure how many ways I can provide such possible parametrizations for the ellipsoid.
Thanks

REPLY [2 votes]: I know this is old but wanted to offer an answer.
If you want to use spherical coordinates, you need to parameterize $r$ as a function of $\theta$ and $\phi$.
Start with
$x = r\cos(\phi)\sin(\theta)$
$y = r\sin(\phi)\sin(\theta)$
$z = r\cos(\theta)$  
Plug into the equation for an ellipsoid and get
$$r = \frac{1}{\sqrt{\left ((\cos(\phi)/a)^2 + (\sin(\phi)/b)^2)\sin(\theta)^2 + (\cos(\theta)/c)^2 \right )}}$$
Given an angle pair $(\theta, \phi)$ the above equation will give you the distance from the center of the ellipsoid to a point on the ellipsoid corresponding to $(\theta, \phi)$.  This may be a little more work than some of the other parameterizations but has its uses.