Editing Stereographic projection (section)

==Properties==
The first stereographic projection defined in the preceding section sends the "south pole" (0,&nbsp;0,&nbsp;−1) of the [[unit sphere]] to (0,&nbsp;0), the equator to the [[unit circle]], the southern hemisphere to the region inside the circle, and the northern hemisphere to the region outside the circle.

The projection is not defined at the projection point {{math|''N''}} = (0,&nbsp;0,&nbsp;1). Small neighborhoods of this point are sent to subsets of the plane far away from (0,&nbsp;0). The closer {{math|''P''}} is to (0,&nbsp;0,&nbsp;1), the more distant its image is from (0,&nbsp;0) in the plane. For this reason it is common to speak of (0,&nbsp;0,&nbsp;1) as mapping to "infinity" in the plane, and of the sphere as completing the plane by adding a [[point at infinity]]. This notion finds utility in [[projective geometry]] and complex analysis. On a merely [[topology|topological]] level, it illustrates how the sphere is [[homeomorphism|homeomorphic]] to the [[one-point compactification]] of the plane.

In [[Cartesian coordinates]] a point {{math|''P''(''x'', ''y'', ''z'')}} on the sphere and its image {{math|''{{prime|P}}''(''X'', ''Y'')}} on the plane either both are [[rational point]]s or none of them:
: <math>P \in \mathbb Q^3 \iff P' \in \mathbb Q^2</math>

[[Image:CartesianStereoProj.png|thumb|left|A Cartesian grid on the plane appears distorted on the sphere. The grid lines are still perpendicular, but the areas of the grid squares shrink as they approach the north pole.]]

[[Image:PolarStereoProj.png|thumb|left|A polar grid on the plane appears distorted on the sphere. The grid curves are still perpendicular, but the areas of the grid sectors shrink as they approach the north pole.]]

Stereographic projection is conformal, meaning that it preserves the angles at which curves cross each other (see figures). On the other hand, stereographic projection does not preserve area; in general, the area of a region of the sphere does not equal the area of its projection onto the plane. The area element is given in {{math|(''X'', ''Y'')}} coordinates by
:<math>dA = \frac{4}{(1 + X^2 + Y^2)^2} \; dX \; dY.</math>
Along the unit circle, where {{math|''X''<sup>2</sup> + ''Y''<sup>2</sup> {{=}} 1}}, there is no inflation of area in the limit, giving a scale factor of 1. Near (0,&nbsp;0) areas are inflated by a factor of 4, and near infinity areas are inflated by arbitrarily small factors.

The metric is given in {{math|(''X'', ''Y'')}} coordinates by
:<math> \frac{4}{(1 + X^2 + Y^2)^2} \; ( dX^2 + dY^2),</math>
and is the unique formula found in [[Bernhard Riemann]]'s ''Habilitationsschrift'' on the foundations of geometry, delivered at Göttingen in 1854, and entitled ''Über die Hypothesen welche der Geometrie zu Grunde liegen''.

No map from the sphere to the plane can be both conformal and area-preserving. If it were, then it would be a local [[isometry]] and would preserve [[Gaussian curvature]]. The sphere and the plane have different Gaussian curvatures, so this is impossible.

[[Circle of a sphere|Circles on the sphere]] that do ''not'' pass through the point of projection are projected to circles on the plane.<ref>{{cite book|last=Ahlfors |first=Lars |author-link=Lars Ahlfors |date=1966 |title=Complex Analysis |url= |location= |publisher=McGraw-Hill, Inc. |page=19 |isbn=}}</ref><ref>{{citation| title=Geometry and the Imagination in Minneapolis| chapter=Stereographic Projection| first1=John| last1=Conway| author-link=John Horton Conway| first2=Peter| last2=Doyle| first3=Jane| last3=Gilman| author3-link=Jane Piore Gilman| first4=Bill| last4=Thurston| author4-link=William Thurston| date=1994-04-12| url=http://www.geom.uiuc.edu/docs/doyle/mpls/handouts/handouts.html| chapter-url=http://www.geom.uiuc.edu/docs/doyle/mpls/handouts/node33.html| publisher=Minnesota University| arxiv=1804.03055| archive-url=https://web.archive.org/web/20210419133952/http://www.geom.uiuc.edu/docs/doyle/mpls/handouts/node33.html| archive-date=2021-04-19| url-status=live| access-date=2022-04-26}}</ref> Circles on the sphere that ''do'' pass through the point of projection are projected to straight lines on the plane. These lines are sometimes thought of as circles through the point at infinity, or circles of infinite radius. These properties can be verified by using the expressions of <math>x,y,z</math> in terms of <math>X, Y, Z,</math> given in {{slink||First formulation}}: using these expressions for a substitution in the equation <math>ax+by+cz-d=0</math> of the plane containing a circle on the sphere, and clearing denominators, one gets the equation of a circle, that is, a second-degree equation with <math>(c-d)(X^2+Y^2)</math> as its quadratic part. The equation becomes linear if <math>c=d,</math> that is, if the plane passes through the point of projection.

All lines in the plane, when transformed to circles on the sphere by the inverse of stereographic projection, meet at the projection point. Parallel lines, which do not intersect in the plane, are transformed to circles tangent at projection point. Intersecting lines are transformed to circles that intersect [[Transversality (mathematics)|transversally]] at two points in the sphere, one of which is the projection point. (Similar remarks hold about the [[real projective plane]], but the intersection relationships are different there.)

[[Image:Riemann Sphere.jpg|right|thumb|175px|The sphere, with various [[loxodrome]]s shown in distinct colors]]
The [[loxodrome]]s of the sphere map to curves on the plane of the form
:<math>R = e^{\Theta/a},\,</math>
where the parameter {{math|''a''}} measures the "tightness" of the loxodrome. Thus loxodromes correspond to [[logarithmic spiral]]s. These spirals intersect radial lines in the plane at equal angles, just as the loxodromes intersect meridians on the sphere at equal angles. <!--Loxodromes may also found by transforming any point with a [[Möbius transformation]], in particular one with a "characteristic constant" that has a nonzero argument and a modulus not equal to one, and which has fixed points that map to diametrically opposite points on the sphere. Continuous iteration may be done by scaling the log of the characteristic constant.-->

{{clear}}
[[File:Inversion by Stereographic.png|thumb]]
The stereographic projection relates to the plane inversion in a simple way. Let {{math|''P''}} and {{math|''Q''}} be two points on the sphere with projections {{math|''{{prime|P}}''}} and {{math|''{{prime|Q}}''}} on the plane. Then {{math|''{{prime|P}}''}} and {{math|''{{prime|Q}}''}} are inversive images of each other in the image of the equatorial circle if and only if {{math|''P''}} and {{math|''Q''}} are reflections of each other in the equatorial plane.

In other words, if:
* {{math|''P''}} is a point on the sphere, but not a 'north pole' {{math|''N''}} and not its [[Antipodal point|antipode]], the 'south pole' {{math|''S''}},
* {{math|''{{prime|P}}''}} is the image of {{math|''P''}} in a stereographic projection with the projection point {{math|''N''}} and
* {{math|''P{{pprime}}''}} is the image of {{math|''P''}} in a stereographic projection with the projection point {{math|''S''}},
then {{math|''{{prime|P}}''}} and {{math|''P{{pprime}}''}} are inversive images of each other in the unit circle.
: <math> \triangle NOP^\prime \sim \triangle P^{\prime\prime}OS \implies OP^\prime:ON = OS : OP^{\prime\prime} \implies OP^\prime \cdot OP^{\prime\prime} = r^2 </math>
{{clear}}