Calculating the position of the 'other' focus point of an orbit

[Chetic]

You know how orbits are made of ellipses and ellipses are made out of two foci?
I'm trying to find a formula where I can extract the position of the other focus point, relative to (0,0).
By the other focus point I mean the one that's generally out in space.

Very much like how the barycenter's position, R, is calculated in this Wikipedia page:
http://en.wikipedia.org/wiki/File:Two-body_Jacobi_coordinates.JPG"

I'm coding a 2d space game and want to predict my orbits using proper math, instead of running the simulation ahead.

P.S: Does this 'other' focus point have a name by the way?

 

[Chetic]

Actually, I think the best way might be to solve for vectors representing periapsis and apoapsis (the highest and the lowest points of orbit).

I have wiki'd myself crazy trying to find some formula for this. Can anybody give me a hint?

Edit: Forgot to mention my problem; I can only find formulas that give the lengths of what would be these two vectors, not any for an actual vector.