## Eccentric Anomaly

The Angle obtained by drawing the Auxiliary Circle of an Ellipse with center and Focus , and drawing a Line Perpendicular to the Semimajor Axis and intersecting it at . The Angle is then defined as illustrated above. Then for an Ellipse with Eccentricity ,

 (1)

But the distance is also given in terms of the distance from the Focus and the Supplement of the Angle from the Semimajor Axis by
 (2)

Equating these two expressions gives
 (3)

which can be solved for to obtain
 (4)

To get in terms of , plug (4) into the equation of the Ellipse
 (5)

 (6)

 (7)

 (8)

Differentiating gives
 (9)

The eccentric anomaly is a very useful concept in orbital mechanics, where it is related to the so-called mean anomaly by Kepler's Equation
 (10)

can also be interpreted as the Area of the shaded region in the above figure (Finch).

Finch, S. Favorite Mathematical Constants.'' http://www.mathsoft.com/asolve/constant/lpc/lpc.html