How do I convert a complex number to a point on the Riemann sphere? Where coordinates on the sphere are $(x,y,z)$ constrained $x^2+y^2+z^2=1$?
There is a formula on Wikipedia going the other direction
$$ \zeta = \frac {x + iy} {1-z} $$
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This is known as "stereographic projection".
In the plane, you can get $(X,Y) = (\frac{x}{1-z},\frac{y}{1-z})$. From the plane to the sphere you get $(x,y,z) = (\frac{2X}{1+X^2+Y^2},\frac{2Y}{1+X^2+Y^2},\frac{X^2+Y^2-1}{1+X^2+Y^2})$.
Then, you just take complex numbers to be $X+iY$.
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1$\begingroup$ There should be a check mark next to the up and down vote buttons. $\endgroup$– BatmanApr 4, 2015 at 2:18