I want to know how to construct the Stereographic projection formulation. I search google and can't find any good resource. Even Wikipedia just give the formula below:
In Cartesian coordinates $(x, y, z)$ on the sphere and $(X, Y)$ on the plane, the projection and its inverse are given by the formulas $$(X,Y)=\left(\frac{x}{1-z},\frac{y}{1-z}\right),$$ $$(x,y,z)=\left(\frac{2X}{1+X^2+Y^2},\frac{2Y}{1+X^2+Y^2},\frac{-1+X^2+Y^2}{1+X^2+Y^2}\right)$$
Can anyone provide me the construction process of this formula or hints?