Describing position on an ellipse as a function of angle

[tomwilliam2]

My textbook (on celestial mechanics) makes a passing reference to position on an ellipse being expressed as:
$r = a(1 - e \cos E)$ before moving on to the substance of the chapter. E is the eccentric anomaly, and r is the distance from the focus to the point on the ellipse.

I'm trying to understand how to derive this expression before moving on. I know that you take the ellipse, with semi-major axis a, and draw an auxiliary circle of radius a around it. Then, the angle opened up as you move around that circle is the eccentric anomaly, E.
Let the foci of the ellipse lie on the x-axis. Now, if $r_x$ is the distance from the centre of the circle to the point on the x-axis which corresponds to the x-component of the position around the auxiliary circle, then:
$\cos E = r_x / a$
because the hypotenuse of the right triangle is the radius, a. I know that $ae$ is the distance from the centre of the circle to the focus of the ellipse. I know that the semi-latus rectum $p=a(1-e^2)$, and this is the distance from the focus to the ellipse directly above the focus.
Now I'm not sure how to proceed from here.
Can anyone point me in the right direction?

 

[Simon Bridge]

Have you had a look see how other people do it?
http://math.etsu.edu/multicalc/prealpha/Chap3/Chap3-2/part4.htm

 

[tomwilliam2]

Thanks! That's the kind of review I'm looking for!