Parametrization for the ellipsoids

Question

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

Answer 1

I think in this way: $x = a\sin(\varphi)\cos(\theta),\quad y = b\sin(\varphi)\sin(\theta),\quad z = c\cos(\varphi)$

Answer 2

You mention polar coordinates, which allow you to parameterise a sphere.

Let $X = \frac{x}{a}$, $Y = \frac{y}{b}$, and $Z = \frac{z}{c}$. Then the equation becomes

$$X^2 + Y^2 + Z^2 = 1.$$

This is the equation of a sphere of radius $1$, so you can parameterise it using polar coordinates. Once you have done that, use the fact that $x = aX$, $y = bY$, and $z = cZ$ to obtain a parameterisation of the ellipsoid.