Spheroid

A spheroid is an Ellipsoid

 (1)

with two Semimajor Axes equal. Orient the Ellipse so that the and axes are equal, then
 (2)

 (3)

where is the equatorial Radius and is the polar Radius. Here is the colatitude, so take to express in terms of latitude.
 (4)

Rewriting gives
 (5)

 (6)

so
 (7)

If , the spheroid is Oblate. If , the spheroid is Prolate. If , the spheroid degenerates to a Sphere.