# Stereographic projection

1. Feb 6, 2005

### T-O7

So i'm trying to prove that the map
$$f(x,y,z) = \frac{(x,y)}{1-z}$$
from the unit sphere S^2 to R^2 is injective by the usual means:
$$f(x_1,y_1,z_1)=f(x_2,y_2,z_2) \Rightarrow (x_1,y_1,z_1)=(x_2,y_2,z_2)$$
But i can't seem to show it....
I end up with the result that
$$\frac{x_1}{x_2}=\frac{y_1}{y_2},\frac{x_1}{x_2}=\frac{1-z_1}{1-z_2}$$,

but i'm uncertain as to what this means for points on a circle......help please?
(i have actually already found the inverse map, but i just found it a little frustrating that i couldn't prove injectiveness just straightforwardly like this..)

2. Feb 7, 2005

### Galileo

Haven't tried it, but since the domain of the function consists of points on the unit sphere, there is a restriction imposed on x,y and z, they cannot have any old values.