Editing Stereographic projection (section)

===First formulation===
[[Image:Stereoprojzero.svg|thumb|right|Stereographic projection of the unit sphere from the north pole onto the plane {{math|''z'' {{=}} 0}}, shown here in [[cross section (geometry)|cross section]]]]

The [[unit sphere]] {{math|''S''<sup>2</sup>}} in three-dimensional space {{math|'''R'''<sup>3</sup>}} is the set of points {{math|(''x'', ''y'', ''z'')}} such that {{math|''x''<sup>2</sup> + ''y''<sup>2</sup> + ''z''<sup>2</sup> {{=}} 1}}. Let {{math|''N'' {{=}} (0,&nbsp;0,&nbsp;1)}} be the "north pole", and let {{mathcal|''M''}}  be the rest of the sphere. The plane {{math|''z'' {{=}} 0}} runs through the center of the sphere; the "equator" is the intersection of the sphere with this plane.

For any point {{math|''P''}} on {{mathcal|''M''}}, there is a unique line through {{math|''N''}} and {{math|''P''}}, and this line intersects the plane {{math|''z'' {{=}} 0}} in exactly one point {{math|''{{prime|P}}''}}, known as the '''stereographic projection''' of {{math|''P''}} onto the plane.

In [[Cartesian coordinates]] {{math|(''x'', ''y'', ''z'')}} on the sphere and {{math|(''X'', ''Y'')}} on the plane, the projection and its inverse are given by the formulas

:<math>\begin{align}(X, Y) &= \left(\frac{x}{1 - z}, \frac{y}{1 - z}\right),\\
(x, y, z) &= \left(\frac{2 X}{1 + X^2 + Y^2}, \frac{2 Y}{1 + X^2 + Y^2}, \frac{-1 + X^2 + Y^2}{1 + X^2 + Y^2}\right).\end{align}</math>

In [[spherical coordinates]] {{math|(''φ'', ''θ'')}} on the sphere (with {{math|''φ''}} the [[zenith angle]], {{math|0 ≤ ''φ'' ≤ π}}, and {{math|''θ''}} the [[azimuth]], {{math|0 ≤ ''θ'' ≤ 2π}}) and [[polar coordinates]] {{math|(''R'', ''Θ'')}} on the plane, the projection and its inverse are

:<math>\begin{align}(R, \Theta) &= \left(\frac{\sin \varphi}{1 - \cos \varphi}, \theta\right) = \left(\cot\frac{\varphi}{2}, \theta\right),\\
(\varphi, \theta) &= \left(2 \arctan \frac{1}{R}, \Theta\right).\end{align}</math>

Here, {{math|''φ''}} is understood to have value {{pi}} when {{math|''R''}} = 0. Also, there are many ways to rewrite these formulas using [[list of trigonometric identities|trigonometric identities]]. In [[cylindrical coordinates]] {{math|(''r'', ''θ'', ''z'')}} on the sphere and polar coordinates {{math|(''R'', ''Θ'')}} on the plane, the projection and its inverse are

:<math>\begin{align}(R, \Theta) &= \left(\frac{r}{1 - z}, \theta\right),\\
(r, \theta, z) &= \left(\frac{2 R}{1 + R^2}, \Theta, \frac{R^2 - 1}{R^2 + 1}\right).\end{align}</math>