We use an ellipsoid to approximate the shape of  the earth.  Four variables are used in the equations for map projections, but only two are independent.  The variables include equatorial and polar radii, flattening, and eccentricity.  The two variable chosen to define an ellipsoid are either a and b, or a and f.  While it makes little practical difference, the computed values will not be exact while the defined variables have exact representations.

a = semi-major (equatorial) axis

b = semi-minor (polar) axis

Flattening:   f = 1 b/a  (often used as 1/f)

Eccentricity: e,  

Going back to the early 19th century, only about 20 ellipsoids have been used for mapping.  Legal and practical issues--all maps and legal property descriptions are based on the datum--preclude changing the ellipsoid and datum every time we obtain a slightly better estimate for the earth's shape. 

Table.  Commonly Used Ellipsoids. (DMA, 1997, Appendix A.1)

Last revision 11/13/2017