These are some images illustrating the orthogonal coordinate system for ellipsoidal surfaces.
Click here to see an interactive ellipsoid.
Ellipsoid octant
Exploded ellipsoid octants
Ellipsoid without sigma correction
Ellipsoid with sigma correction
Ellipsoid plus stereoscopic projection
Stereoscopic projection
Torus mapped to plane. This is the image of an elliptic function and looks suspiciously like the stereographic projection of an ellipsoid.
Ellipsoidal texture mapping