Editing Stereographic projection (section)

==Definition==

===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>

===Other conventions===
[[Image:Stereoprojnegone.svg|thumb|left|Stereographic projection of the unit sphere from the north pole onto the plane {{math|''z'' {{=}} −1}}, shown here in cross section]]
Some authors<ref>Cf. Apostol (1974) p. 17.</ref> define stereographic projection from the north pole (0,&nbsp;0,&nbsp;1) onto the plane {{math|''z'' {{=}} −1}}, which is tangent to the unit sphere at the south pole (0,&nbsp;0,&nbsp;−1). This can be described as a [[Function composition|composition]] of a projection onto the equatorial plane described above, and a [[homothety]] from it to the polar plane. The homothety scales the image by a factor of 2 (a ratio of a diameter to a radius of the sphere), hence the values {{math|''X''}} and {{math|''Y''}} produced by this projection are exactly twice those produced by the equatorial projection described in the preceding section. For example, this projection sends the equator to the circle of radius 2 centered at the origin. While the equatorial projection produces no infinitesimal area distortion along the equator, this pole-tangent projection instead produces no infinitesimal area distortion at the south pole.

Other authors<ref name=Gelfand_M_S>{{harvnb|Gelfand|Minlos|Shapiro|1963}}</ref> use a sphere of radius {{math|{{sfrac|1|2}}}} and the plane {{math|''z'' {{=}} −{{sfrac|1|2}}}}. In this case the formulae become
:<math>\begin{align}(x,y,z) \rightarrow (\xi, \eta) &= \left(\frac{x}{\frac{1}{2} - z}, \frac{y}{\frac{1}{2} - z}\right),\\
(\xi, \eta) \rightarrow (x,y,z) &= \left(\frac{\xi}{1 + \xi^2 + \eta^2}, \frac{\eta}{1 + \xi^2 + \eta^2}, \frac{-1 + \xi^2 + \eta^2}{2 + 2\xi^2 + 2\eta^2}\right).\end{align}</math>

[[Image:StereographicGeneric.svg|thumb|right|Stereographic projection of a sphere from a point {{math|''Q''}} onto the plane {{math|''E''}}, shown here in cross section]]
In general, one can define a stereographic projection from any point {{math|''Q''}} on the sphere onto any plane {{math|''E''}} such that
*{{math|''E''}} is perpendicular to the diameter through {{math|''Q''}}, and
*{{math|''E''}} does not contain {{math|''Q''}}.
As long as {{math|''E''}} meets these conditions, then for any point {{math|''P''}} other than {{math|''Q''}} the line through {{math|''P''}} and {{math|''Q''}} meets {{math|''E''}} in exactly one point {{math|''{{prime|P}}''}}, which is defined to be the stereographic projection of ''P'' onto ''E''.<ref>Cf. Pedoe (1988).</ref>

===Generalizations===
More generally, stereographic projection may be applied to the unit [[n-sphere|{{math|''n''}}-sphere]] {{math|''S''<sup>''n''</sup>}} in ({{math|''n'' + 1}})-dimensional [[Euclidean space]] {{math|'''E'''<sup>''n''+1</sup>}}.  If {{math|''Q''}} is a point of {{math|''S''<sup>''n''</sup>}} and {{math|''E''}} a [[hyperplane]] in {{math|'''E'''<sup>''n''+1</sup>}}, then the stereographic projection of a point {{math|''P'' ∈ ''S''<sup>''n''</sup> − {{mset|''Q''}}}} is the point {{math|''{{prime|P}}''}} of intersection of the line {{math|{{overline|''QP''}}}} with {{math|''E''}}. In [[Cartesian coordinates]] ({{math|''x''{{sub|''i''}}}}, {{math|''i''}} from 0 to {{math|''n''}}) on {{math|''S''<sup>''n''</sup>}} and ({{math|''X''{{sub|''i''}}}}, {{math|''i''}} from 1 to ''n'') on {{math|''E''}}, the projection from {{math|1=''Q'' = (1, 0, 0, ..., 0) ∈ ''S''<sup>''n''</sup>}} is given by
<math display="block">X_i = \frac{x_i}{1 - x_0} \quad (i = 1, \dots, n).</math>
Defining
<math display="block">s^2=\sum_{j=1}^n X_j^2 = \frac{1+x_0}{1-x_0},</math>
the inverse is given by
<math display="block">x_0 = \frac{s^2-1}{s^2+1} \quad \text{and} \quad x_i = \frac{2 X_i}{s^2+1} \quad (i = 1, \dots, n).</math>

Still more generally, suppose that {{math|''S''}} is a (nonsingular) [[quadric|quadric hypersurface]] in the [[projective space]] {{math|'''P'''<sup>''n''+1</sup>}}. In other words, {{math|''S''}} is the locus of zeros of a non-singular quadratic form {{math|''f''(''x''<sub>0</sub>, ..., ''x''<sub>''n''+1</sub>)}} in the [[homogeneous coordinates]] {{math|''x''<sub>''i''</sub>}}. Fix any point {{math|''Q''}} on {{math|''S''}} and a hyperplane {{math|''E''}} in {{math|'''P'''<sup>''n''+1</sup>}} not containing {{math|''Q''}}.  Then the stereographic projection of a point {{math|''P''}} in {{math|''S'' − {{mset|''Q''}}}} is the unique point of intersection of {{math|{{overline|''QP''}}}} with {{math|''E''}}.  As before, the stereographic projection is conformal and invertible on a non-empty Zariski open set. The stereographic projection presents the quadric hypersurface as a [[rational variety|rational hypersurface]].<ref>Cf. Shafarevich (1995).</ref>  This construction plays a role in [[algebraic geometry]] and [[conformal geometry]].