Editing Stereographic projection (section)

===Complex analysis===
[[File:Riemann sphere1.svg|thumb|300px|right|The stereographic projection from the North pole of a sphere to its equatorial plane establishes a [[one to one correspondence]] between the sphere and the equatorial plane extended with a [[point at infinity]] denoted {{math|∞}}. When the equatorial plane is the [[complex plane]], this provides a visualization of the [[Riemann sphere]]]]

Although any stereographic projection misses one point on the sphere (the projection point), the entire sphere can be mapped using two projections from distinct projection points. In other words, the sphere can be covered by two stereographic [[Parametrization (geometry)|parametrization]]s (the inverses of the projections) from the plane. The parametrizations can be chosen to induce the same [[orientation (space)|orientation]] on the sphere. Together, they describe the sphere as an oriented [[Surface (topology)|surface]] (or two-dimensional [[manifold]]).

This construction has special significance in complex analysis. The point {{math|(''X'', ''Y'')}} in the real plane can be identified with the [[complex number]] {{math|''ζ'' {{=}} ''X'' + i''Y''}}. The stereographic projection from the north pole onto the equatorial plane is then
:<math>\begin{align} \zeta &= \frac{x + i y}{1 - z},\\
\\
(x, y, z) &= \left(\frac{2 \operatorname{Re} \zeta}{1 + \bar \zeta \zeta}, \frac{2 \operatorname{Im} \zeta}{1 + \bar \zeta \zeta}, \frac{-1 + \bar \zeta \zeta}{1 + \bar \zeta \zeta}\right).\end{align}</math>

Similarly, letting {{math|''ξ'' {{=}} ''X'' − i''Y''}} be another complex coordinate, the functions
:<math>\begin{align} \xi &= \frac{x - i y}{1 + z},\\
(x, y, z) &= \left(\frac{2 \operatorname{Re} \xi}{1 + \bar \xi \xi}, \frac{-2 \operatorname{Im} \xi}{1 + \bar \xi \xi}, \frac{1 - \bar \xi \xi}{1 + \bar \xi \xi}\right)\end{align}</math>
define a stereographic projection from the south pole onto the equatorial plane. The transition maps between the {{math|''ζ''}}- and {{math|''ξ''}}-coordinates are then {{math|''ζ'' {{=}} {{sfrac|1|''ξ''}}}} and {{math|''ξ'' {{=}} {{sfrac|1|''ζ''}}}}, with {{math|''ζ''}} approaching 0 as {{math|''ξ''}} goes to infinity, and ''vice versa''. This facilitates an elegant and useful notion of infinity for the complex numbers and indeed an entire theory of [[meromorphic function]]s mapping to the [[Riemann sphere]]. The [[Riemannian_manifold#Riemannian_metrics_and_Riemannian_manifolds|standard metric]] on the unit sphere agrees with the [[Fubini–Study metric]] on the Riemann sphere.