Inverse map

1. Sep 16, 2009

sk1001

1. The problem statement, all variables and given/known data

Let S = {(x,y,z) be an element of R3: x2+y2+z2=1} be the unit sphere

I have a map that takes coordinates from the xy plane to points on the sphere.
This map is given by:

σ:R2->S\{(0,0,1)}
σ(u,v) = (2u, 2v, u2+v2-1)/1+u2+v2

Now I wish to find the inverse map:

σ-1(u,v):S\{(0,0,1)}->R2

that is, to take points on the sphere onto the xy-plane (the stereographic projection)

3. The attempt at a solution

I'm really not quite sure where to start. I understand the logic behind what is asked, but mathematically, I don't quite know!