Ellipsoid - Parameterization

Parameterization

The surface of the ellipsoid may be parameterized in several ways. One possible choice which singles out the 'z'-axis is:

$\begin{align} x&=a\,\cos u\cos v,\\ y&=b\,\cos u\sin v,\\ z&=c\,\sin u;\end{align}\,\!$

where

$-{\pi}/{2}\leq u\leq+{\pi}/{2}, \qquad -\pi\leq v\leq+\pi.\!\,\!$

The parameters may be interpreted as spherical coordinates. For constant u, that is on the ellipse which is the intercept with a constant z plane, v then plays the role of the eccentric anomaly for that ellipse. For constant v on a plane through the Oz axis the parameter u plays the same role for the ellipse of intersection. Two other similar parameterizations are possible, each with their own interpretations. Only on an ellipse of revolution can a unique definition of reduced latitude be made.