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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.


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For ellipsis it looks like this $x = a\cos(\varphi), y = b\sin(\varphi)$ –  Nikita Evseev Oct 2 '12 at 10:05

3 Answers 3

The answer to your question will depend on what you want to do with the ellipsoid. The Wikipedia page on geodesics on ellipsoids gives three possible parametrizations of the surface: (1) geographic latitude and longitude (useful if you're determining your position by astronomical observations); (2) parametric coordinates, probably the simplest to deal with computationally; (3) ellipsoidal coordinates, nice in that they are lines of curvature and hence orthogonal.

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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.

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

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