Denote the longitude minus the central meridian by s. Denote vectors by covering arrow like so . Denote points in 3-dimensional space by lower case . Denote points on the map by upper case . Let the origin of our 3 dimensional coordinate system be at the center of the earth. Let the z axis lie along the earths axis and up through the north pole. Let the x axis go from the center of the earth along the equatorial plane to the central meridian. The y axis also lies in the equatorial plane and is perpendicular to and east of the x axis.
The parametric equations of the spheroid is :-
Where t is the geocentirc latitude and s is the longitude.