Falling short
The polar form of the equation for the ellipse can be derived in a number of ways.
If we consider the ellipse the locus of points which maintain the sum of distances between from point to foci constant, then consider the following diagram:
The red point and the cross are the two foci. is the length of the major axis.
From the definition: .
From Pythagras theorem,
Therefore,
using
Simplifying and solving for yields:
Eccentricity, ,
Note that when , . at this point is the semiminor axis. Let represent the semiminor axis. As ,
From the above graphic, the periapsis occurs when , yielding . Similarly, the apoapsis occurs when and .
The ellipse can be defined (as can all conic sections) as the locus of points that maintain a constant ratio of distance from a fixed line (directrix) to distance to a fixed point (focus). Using the same ellipse as above and the same angle convention (as illustrated in the following graphic) the derivation is straightforward. Let be the distance between the focus and the directrix, and emerges as illustrated.
Solving for ,
This is consistent with the above derivation.
Comment
 The graphics were made using Mathematica and the drawing tool.
 The specific ellipse used was where . The arguments are, however, general.

July 10, 2011 at 5:40 amAnalytical Geometry  salesmarketingessentials