Editing Stereographic projection (section)

===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>