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 (◇) into the equation of the ellipse
 |
(5)
|
Rearranging,
 |
(6)
|
and plugging in (◇) then gives
 |  |  |
(7)
|
 |  |  |
(8)
|
Solving for 
gives
 |
(9)
|
so differentiating yields the result
 |
(10)
|
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
 |
(11)
|
can also be interpreted as the area of the shaded region
in the above figure (Finch 2003).
