View source for Stereographic map projection

{{Short description|Type of conformal map projection}}
[[File:Stereographic projection SW.JPG|250px|thumb|Stereographic projection of the world north of 30°S. 15° graticule.]]
[[File:Stereographic with Tissot's Indicatrices of Distortion.svg|250px|thumb|The stereographic projection with [[Tissot's indicatrix]] of deformation.]]

The '''stereographic projection''', also known as the '''planisphere projection''' or the '''azimuthal conformal projection''', is a [[conformal map projection]] whose use dates back to antiquity. Like the [[orthographic projection in cartography|orthographic projection]] and [[gnomonic projection]], the [[stereographic projection]] is an [[Map projection#Azimuthal .28projections onto a plane.29|azimuthal projection]], and when on a sphere, also a [[General Perspective projection|perspective projection]].

On an [[ellipsoid]], the perspective definition of the stereographic projection is not conformal, and adjustments must be made to preserve its azimuthal and conformal properties. The [[universal polar stereographic coordinate system]] uses one such ellipsoidal implementation.

== History ==

[[Image:Mercator World Map.jpg|200px|thumb|World map made by Rumold Mercator in 1587, using two equatorial aspects of the stereographic projection.]]

The stereographic projection was likely known in its polar aspect to the [[ancient Egypt]]ians, though its invention is often credited to [[Hipparchus]], who was the first Greek to use it.{{fact|date=September 2021}} Its oblique aspect was used by Greek Mathematician [[Theon of Alexandria]] in the fourth century, and its equatorial aspect was used by Arab astronomer [[Al-Zarkali]] in the eleventh century. The earliest written description of it is Ptolemy's ''[[Planisphaerium]]'', which calls it the "planisphere projection".

The stereographic projection was exclusively used for star charts until 1507, when Walther Ludd of St. Dié, Lorraine created the first known instance of a stereographic projection of the Earth's surface. Its popularity in cartography increased after [[Rumold Mercator]] used its equatorial aspect for his 1595 atlas.<ref name="Working manual">Snyder, John P. 1987. "Map Projections---A Working Manual". ''Professional Paper''. United States Geological Survey. 1395: 154--163. {{ISBN|0-226-76746-9}}.</ref> It subsequently saw frequent use throughout the seventeenth century with its equatorial aspect being used for maps of the [[Eastern hemisphere|Eastern]] and [[Western hemisphere]]s.<ref name="Flattening" />

In 1695, [[Edmond Halley]], motivated by his interest in [[star chart]]s, published the first mathematical proof that this map is [[Conformal map|conformal]].<ref>Timothy Feeman. 2002. "Portraits of the Earth: A Mathematician Looks at Maps". American Mathematical Society.</ref> He used the recently established tools of [[calculus]], invented by his friend [[Isaac Newton]].

== Formulae ==

The spherical form of the stereographic projection is usually expressed in polar coordinates:

:<math>
\begin{align}
  r &= 2 R \tan\left(\frac{\pi}{4} - \frac{\varphi}{2}\right) \\
  \theta &= \lambda
\end{align}
</math>

where <math>R</math> is the radius of the sphere, and <math>\varphi</math> and <math>\lambda</math> are the latitude and longitude, respectively.

The [[Figure of the Earth#Sphere|sphere]] is normally chosen to model the Earth when the extent of the mapped region exceeds a few hundred kilometers in length in both dimensions. For maps of smaller regions, an [[Figure of the Earth#Ellipsoid of revolution|ellipsoidal model]] must be chosen if greater accuracy is required.<ref name="Working manual" />

The ellipsoidal form of the polar ellipsoidal projection uses [[conformal latitude]]. There are various forms of transverse or oblique stereographic projections of ellipsoids. One method uses double projection via a conformal sphere, while other methods do not.

Examples of transverse or oblique stereographic projections include the Miller Oblated Stereographic<ref>{{Cite journal |doi = 10.1559/152304086783899908|title = The Miller Oblated Stereographic Projection for Africa, Europe, Asia and Australasia|journal = The American Cartographer|volume = 13|issue = 3|pages = 253–261|year = 1986|last1 = Sprinsky|first1 = William H.|last2 = Snyder|first2 = John P.}}</ref> and the [[Roussilhe oblique stereographic projection]].<ref name="Flattening">Snyder, John P. (1993). ''Flattening the Earth: Two Thousand Years of Map Projections'' p.~169. Chicago and London: The University of Chicago Press. {{ISBN|0-226-76746-9}}.</ref>

== Properties ==

As an azimuthal projection, the stereographic projection faithfully represents the relative directions of all [[great circle]]s passing through its center point. As a conformal projection, it faithfully represents angles everywhere. In addition, in its spherical form, the stereographic projection is the only map projection that renders all [[Circle of a sphere|small circles]] as circles.

[[Image:Stereographic projection in 3D.svg|200px|thumb|3D illustration of the geometric construction of the stereographic projection.]]

The spherical form of the stereographic projection is equivalent to a perspective projection where the point of perspective is on the point on the globe opposite the center point of the map.

Because the expression for <math>r</math> diverges as <math>\varphi</math> approaches <math>-\frac{\pi}{2}</math>, the stereographic projection is infinitely large, and showing the South Pole (for a map centered on the North Pole) is impossible. However, it is possible to show points arbitrarily close to the South Pole as long as the boundaries of the map are extended far enough.<ref name="Working manual" />

== Derived projections ==

The parallels on the [[Gall stereographic projection]] are distributed with the same spacing as those on the central meridian of the [[Map projection#Aspect of the projection|transverse]] stereographic projection.

The [[GS50 projection]] is formed by mapping the [[Map projection#Aspect of the projection|oblique]] stereographic projection to the [[complex plane]] and then transforming points on it via a tenth-order polynomial.

{{comparison_azimuthal_projections.svg}}

==References==
{{Reflist}}

{{Map projections}}

{{Authority control}}

[[Category:Conformal projections]]