Editing Stereographic projection (section)

==Applications within mathematics==

===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.

===Visualization of lines and planes===
[[Image:Sfsp111.gif|thumb|right|Animation of [[Kikuchi line (solid state physics)|Kikuchi lines]] of four of the eight <111> zones in an fcc crystal. Planes edge-on (banded lines) intersect at fixed angles.]]

The set of all lines through the origin in three-dimensional space forms a space called the [[real projective plane]]. This plane is difficult to visualize, because it cannot be [[Embedding|embedded]] in three-dimensional space.

However, one can visualize it as a disk, as follows. Any line through the origin intersects the southern hemisphere {{math|''z''}}&nbsp;≤&nbsp;0 in a point, which can then be stereographically projected to a point on a disk in the XY plane. Horizontal lines through the origin intersect the southern hemisphere in two [[antipodal point]]s along the equator, which project to the boundary of the disk. Either of the two projected points can be considered part of the disk; it is understood that antipodal points on the equator represent a single line in 3 space and a single point on the boundary of the projected disk (see [[quotient topology]]). So any set of lines through the origin can be pictured as a set of points in the projected disk.  But the boundary points behave differently from the boundary points of an ordinary 2-dimensional disk, in that any one of them is simultaneously close to interior points on opposite sides of the disk (just as two nearly horizontal lines through the origin can project to points on opposite sides of the disk).

Also, every plane through the origin intersects the unit sphere in a great circle, called the ''trace'' of the plane. This circle maps to a circle under stereographic projection. So the projection lets us visualize planes as circular arcs in the disk.  Prior to the availability of computers, stereographic projections with great circles often involved drawing large-radius arcs that required use of a [[beam compass]].  Computers now make this task much easier.

Further associated with each plane is a unique line, called the plane's ''pole'', that passes through the origin and is perpendicular to the plane. This line can be plotted as a point on the disk just as any line through the origin can. So the stereographic projection also lets us visualize planes as points in the disk. For plots involving many planes, plotting their poles produces a less-cluttered picture than plotting their traces.

This construction is used to visualize directional data in crystallography and geology, as described below.

===Other visualization===
Stereographic projection is also applied to the visualization of [[polytope]]s. In a [[Schlegel diagram]], an {{math|''n''}}-dimensional polytope in {{math|'''R'''<sup>''n''+1</sup>}} is projected onto an {{math|''n''}}-dimensional sphere, which is then stereographically projected onto {{math|'''R'''<sup>''n''</sup>}}. The reduction from {{math|'''R'''<sup>''n''+1</sup>}} to {{math|'''R'''<sup>''n''</sup>}} can make the polytope easier to visualize and understand.

===Arithmetic geometry===
[[Image:Stereographic projection of rational points.svg|thumb|right|The [[rational point]]s on a circle correspond, under stereographic projection, to the rational points of the line.]]
In elementary [[arithmetic geometry]], stereographic projection from the unit circle provides a means to describe all primitive [[Pythagorean triple]]s.  Specifically, stereographic projection from the north pole (0,1) onto the {{math|''x''}}-axis gives a one-to-one correspondence between the [[rational number]] points {{math|(''x'', ''y'')}} on the unit circle (with {{math|''y'' ≠ 1}}) and the [[rational point]]s of the {{math|''x''}}-axis.  If {{math|({{sfrac|''m''|''n''}}, 0)}} is a rational point on the {{math|''x''}}-axis, then its inverse stereographic projection is the point

:<math>\left(\frac{2mn}{m^2+n^2}, \frac{m^2-n^2}{m^2+n^2}\right)</math>

which gives Euclid's formula for a Pythagorean triple.

===Tangent half-angle substitution===
{{main|Tangent half-angle substitution}}
[[File:WeierstrassSubstitution.svg|thumb|right]]
The pair of trigonometric functions {{math|(sin ''x'', cos ''x'')}} can be thought of as parametrizing the unit circle.  The stereographic projection gives an alternative parametrization of the unit circle:
:<math>\cos x = \frac{1 - t^2}{1 + t^2},\quad \sin x = \frac{2 t}{t^2 + 1}.</math>
Under this reparametrization, the length element {{math|''dx''}} of the unit circle goes over to
:<math>dx = \frac{2 \, dt}{t^2 + 1}.</math>
This substitution can sometimes simplify [[integral]]s involving trigonometric functions.