Can Stereographic Projection from Unit Sphere to Plane be Proven as Injective?

[T-O7]

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

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..)

 

[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.

 

[M98Ranger]

Stereographic projection is a geometric technique used to map points on a sphere to points on a plane. In this case, the unit sphere S^2 is being mapped to the plane R^2 using the function f(x,y,z) = \frac{(x,y)}{1-z}. The goal is to prove that this mapping is injective, meaning that each point on the sphere is uniquely mapped to a point on the plane.

To prove injectivity, the usual method is to assume that two points on the sphere, (x_1,y_1,z_1) and (x_2,y_2,z_2), are mapped to the same point on the plane, f(x_1,y_1,z_1)=f(x_2,y_2,z_2). This would mean that the two points have the same coordinates, (x_1,y_1) = (x_2,y_2), and also that the ratio of their z-coordinates is equal, \frac{1-z_1}{1-z_2} = 1. However, as you have found, this does not necessarily mean that the two points are the same, since there are multiple points on the sphere that can have the same projection onto the plane.

In order to show that the mapping is injective, you need to consider the entire mapping function, not just the coordinates. One way to do this is to find the inverse mapping, which you have already done. This means that for each point on the plane, there is a unique point on the sphere that is mapped to it, and vice versa. This shows that the mapping is one-to-one and therefore injective.

It is important to note that the mapping from the sphere to the plane is not an isomorphism, meaning that the two structures are not exactly the same. This is why the usual method of proving injectivity does not work in this case.

In conclusion, while it may be frustrating that the usual method of proving injectivity does not work for stereographic projection, it is important to understand the underlying principles and consider the entire mapping function in order to prove injectivity.