A coordinate system where the third dimension is not just any dimension, but a dimension of pure, unadulterated spherical madness.
In the ellipsoidal system, we use three coordinates: x, y, and theta. Theta is the angle around the equator, x and y are the usual Cartesian axes, but with a twist: they're both functions of theta. The result is a coordinate system where every point is on the surface of a sphere. Or, you know, not really.
For example, if x = cos(theta) and y = sin(theta), then every point is on the surface of a unit circle. But what about if x = sin(theta) and y = cos(theta)? Now we're on the surface of a unit circle, but with a twist! It's like the difference between a regular circle and a torus, but, like, in 3D.
If you're working with spheres, you might want to use the ellipsoidal system. It's like a more interesting, more complex, more... well, more ellipsoidal version of Cartesian coordinates. And who doesn't love a good challenge to their math skills?
Ellipsoidal Coordinates for Spheres Ellipsoidal Coordinates for Torus
Using Ellipsoidal Coordinates in Practice Ellipsoidal Coordinates in the Real World