Holomorphic stereographic atlas of the sphere
= 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.