= Holomorphic stereographic atlas of the sphere
{title2=$\eta=1/\zeta$}
On the unit <sphere>, use $\zeta=(x+iy)/(1-z)$ away from the north pole and $\eta=(x-iy)/(1+z)$ away from the south pole. Their overlap relation $\eta=1/\zeta$ is a <holomorphic map>, giving a <holomorphic atlas> for the <Riemann sphere>. The opposite sign of $iy$ in the second <manifold chart> ensures a holomorphic rather than antiholomorphic transition.
Writing the sphere coordinates as $(X,Y,Z)$ avoids confusing their height with the <complex coordinate> $z$. The two <manifold charts> cover the whole <unit sphere>:
$$
z=\frac{X+iY}{1-Z}\quad\text{on }S^2\setminus\{N\},\qquad w=\frac{X-iY}{1+Z}\quad\text{on }S^2\setminus\{S\}.
$$
On their overlap, $X^2+Y^2=1-Z^2$ gives
$$
zw=\frac{X^2+Y^2}{(1-Z)(1+Z)}=1,\qquad w=\frac1z,\qquad \frac{dw}{dz}=-\frac1{z^2}\ne0.
$$
The <coordinate transformation> and its inverse are <holomorphic>, so these <manifold charts> make the <sphere> a <Riemann surface>. The sign in $X-iY$ is essential. Geometric projection from the south pole with the same planar axis labels would give $v=(X+iY)/(1+Z)=1/\overline z$. The <complex conjugation> in that transition prevents it from being <holomorphic>. For $z=i$, the correct second coordinate is $w=-i$: positive $Y$ corresponds to negative <imaginary part> in this <manifold chart>.
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