Editing Stereographic projection (section)

==Applications to other disciplines==

===Cartography===
{{main|Stereographic map projection}}

The fundamental problem of cartography is that no map from the sphere to the plane can accurately represent both angles and areas. In general, area-preserving [[map projection]]s are preferred for [[statistics|statistical]] applications, while angle-preserving (conformal) map projections are preferred for [[navigation]].

Stereographic projection falls into the second category. When the projection is centered at the Earth's north or south pole, it has additional desirable properties: It sends [[Meridian (geography)|meridian]]s to rays emanating from the origin and [[Circle of latitude|parallel]]s to circles centered at the origin.

<gallery class="center" widths="250px" heights="250px">
File:Stereographic projection SW.JPG|Stereographic projection of the world north of 30°S. 15° graticule.
File:Stereographic with Tissot's Indicatrices of Distortion.svg|The stereographic projection with [[Tissot's indicatrix]] of deformation.
</gallery>

===Planetary science===
[[File:LRO WAC North Pole Mosaic (PIA14024).jpg|thumb|left|A stereographic projection of the [[Moon]], showing regions polewards of 60° North. Craters which are [[Circle of a sphere|circles on the sphere]] appear circular in this projection, regardless of whether they are close to the pole or the edge of the map.]]
The stereographic is the only projection that maps all [[circle of a sphere|circles on a sphere]] to [[Circle|circles on a plane]]. This property is valuable in planetary mapping where craters are typical features. The set of circles passing through the point of projection have unbounded radius, and therefore [[Degeneracy (mathematics)|degenerate]] into lines.
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===Crystallography===
[[Image:DiamondPoleFigure111.png|right|thumb|A crystallographic pole figure for the [[Diamond cubic|diamond lattice]] in [[Miller index|[111] direction]]]]
{{main|Pole figure}}

In [[crystallography]], the orientations of [[crystal]] axes and faces in three-dimensional space are a central geometric concern, for example in the interpretation of [[X-ray diffraction|X-ray]] and [[electron diffraction]] patterns. These orientations can be visualized as in the section [[Stereographic projection#Visualization of lines and planes|Visualization of lines and planes]] above. That is, crystal axes and poles to crystal planes are intersected with the northern hemisphere and then plotted using stereographic projection. A plot of poles is called a '''pole figure'''.

In [[electron diffraction]], [[Kikuchi line (solid state physics)|Kikuchi line]] pairs appear as bands decorating the intersection between lattice plane traces and the [[Ewald sphere]] thus providing ''experimental access'' to a crystal's stereographic projection.  Model Kikuchi maps in reciprocal space,<ref>M. von Heimendahl, W. Bell and G. Thomas (1964) Applications of Kikuchi line analyses in electron microscopy, ''J. Appl. Phys.'' '''35''':12, 3614&ndash;3616.</ref> and fringe visibility maps for use with bend contours in direct space,<ref>P. Fraundorf, Wentao Qin, P. Moeck and Eric Mandell (2005) Making sense of nanocrystal lattice fringes, ''J. Appl. Phys.'' '''98''':114308.</ref> thus act as road maps for exploring orientation space with crystals in the [[transmission electron microscope]].
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===Geology===
[[File:Stero projection structural geology.png|thumb|300px|left|Use of lower hemisphere stereographic projection to plot planar and linear data in structural geology, using the example of a fault plane with a slickenside lineation]]

Researchers in [[structural geology]] are concerned with the orientations of planes and lines for a number of reasons. The [[Foliation (geology)|foliation]] of a rock is a planar feature that often contains a linear feature called [[Lineation (geology)|lineation]]. Similarly, a [[Fault (geology)|fault]] plane is a planar feature that may contain linear features such as [[slickenside]]s.

These orientations of lines and planes at various scales can be plotted using the methods of the [[Stereographic projection#Visualization of lines and planes|Visualization of lines and planes]] section above. As in crystallography, planes are typically plotted by their poles. Unlike crystallography, the southern hemisphere is used instead of the northern one (because the geological features in question lie below the Earth's surface). In this context the stereographic projection is often referred to as the '''equal-angle lower-hemisphere projection'''. The equal-area lower-hemisphere projection defined by the [[Lambert azimuthal equal-area projection]] is also used, especially when the plot is to be subjected to subsequent statistical analysis such as density [[contour line|contouring]].<ref name="Leyshon _&_Lisle_2004">{{Cite book |last1=Lisle |first1=R.J. |title=Stereographic Projection Techniques for Geologists and Civil Engineers |last2=Leyshon |first2=P.R. |publisher=Cambridge University Press |year=2004 |isbn=9780521535823 |edition=2}}</ref>
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===Rock mechanics===
The stereographic projection is one of the most widely used methods for evaluating rock slope stability. It allows for the representation and analysis of three-dimensional orientation data in two dimensions. Kinematic analysis within stereographic projection is used to assess the potential for various modes of rock slope failures—such as plane, wedge, and toppling failures—which occur due to the presence of unfavorably oriented discontinuities.<ref name=":0">{{Cite book |last1=Hoek |first1=Evert |url=https://books.google.com/books?id=VSccRRWxHf4C&dq=hoek+rock+slope+engineering&pg=PA7 |title=Rock Slope Engineering: Third Edition |last2=Bray |first2=Jonathan D. |date=1981-06-30 |publisher=CRC Press |isbn=978-0-419-16010-6 }}</ref><ref>{{Cite book |last1=Lisle |first1=Richard J. |url=https://www.cambridge.org/core/books/stereographic-projection-techniques-for-geologists-and-civil-engineers/3462D8306346A41ADE5667A3B0A3E82F |title=Stereographic Projection Techniques for Geologists and Civil Engineers |last2=Leyshon |first2=Peter R. |date=2004 |publisher=Cambridge University Press |isbn=978-0-521-53582-3 |edition=2 |location=Cambridge |doi=10.1017/cbo9781139171366}}</ref> This technique is particularly useful for visualizing the orientation of rock slopes in relation to discontinuity sets, facilitating the assessment of the most likely failure type.<ref name=":0" /> For instance, plane failure is more likely when the strike of a discontinuity set is parallel to the slope, and the discontinuities dip towards the slope at an angle steep enough to allow sliding, but not steeper than the slope itself.

Additionally, some authors have developed graphical methods based on stereographic projection to easily calculate geometrical correction parameters—such as those related to the parallelism between the slope and discontinuities, the dip of the discontinuity, and the relative angle between the discontinuity and the slope—for rock mass classifications in slopes, including [[slope mass rating]] (SMR)<ref>{{Cite journal |last1=Tomás |first1=R. |last2=Cuenca |first2=A. |last3=Cano |first3=M. |last4=García-Barba |first4=J. |date=2012-01-04 |title=A graphical approach for slope mass rating (SMR) |url=https://www.sciencedirect.com/science/article/abs/pii/S0013795211002572 |journal=Engineering Geology |volume=124 |pages=67–76 |doi=10.1016/j.enggeo.2011.10.004 |bibcode=2012EngGe.124...67T |issn=0013-7952}}</ref> and [[rock mass rating]].<ref>{{Cite journal |last1=Moon |first1=Vicki |last2=Russell |first2=Geoff |last3=Stewart |first3=Meagan |date=July 2001 |title=The value of rock mass classification systems for weak rock masses: a case example from Huntly, New Zealand |url=http://dx.doi.org/10.1016/s0013-7952(01)00024-2 |journal=Engineering Geology |volume=61 |issue=1 |pages=53–67 |doi=10.1016/s0013-7952(01)00024-2 |bibcode=2001EngGe..61...53M |issn=0013-7952}}</ref>

===Photography===
[[File:Cmglee_Wikimania2016_Esino_Lario_Last_Supper_tinyplanet.jpg|thumb|left|250px|Stereographic projection of the spherical panorama of the Last Supper sculpture by [[Michele Vedani]] in [[Esino Lario]], Lombardy, Italy during [[Wikimania 2016]]]]
[[File:Vue circulaire des montagnes qu ‘on decouvre du sommet du Glacier de Buet, from Horace-Benedict de Saussure, Voyage dans les Alpes, précédés d'un essai sur l'histoire naturelle des environs de Geneve. Neuchatel, l779-96, pl. 8.jpg|thumb|right|"Vue circulaire des montagnes qu'on découvre du sommet du Glacier de Buet", Horace-Benedict de Saussure, ''Voyage dans les Alpes, précédés d'un essai sur l'histoire naturelle des environs de Geneve''. Neuchatel, 1779–96, pl. 8.]]
Some [[fisheye lens]]es use a stereographic projection to capture a wide-angle view.<ref>[http://www.syopt.co.kr/eng/product/8mm.asp Samyang 8 mm {{f/}}3.5 Fisheye CS] {{Webarchive|url=https://web.archive.org/web/20110629031236/http://www.syopt.co.kr/eng/product/8mm.asp |date=2011-06-29 }}</ref> Compared to more traditional fisheye lenses which use an equal-area projection, areas close to the edge retain their shape, and straight lines are less curved. However, stereographic fisheye lenses are typically more expensive to manufacture.<ref>{{cite web|url=http://www.lenstip.com/160.1-Lens_review-Samyang_8_mm_f_3.5_Aspherical_IF_MC_Fish-eye-Introduction.html|title=Samyang 8 mm f/3.5 Aspherical IF MC Fish-eye|publisher=lenstip.com|access-date=2011-07-07}}</ref>  Image remapping software, such as [[Panotools]], allows the automatic remapping of photos from an equal-area fisheye to a stereographic projection.

The stereographic projection has been used to map spherical [[panorama]]s, starting with [[Horace Bénédict de Saussure]]'s in 1779. This results in effects known as a ''little planet'' (when the center of projection is the [[nadir]]) and a ''tube'' (when the center of projection is the [[zenith]]).<ref name="German2007">German ''et al.'' (2007).</ref>

The popularity of using stereographic projections to map panoramas over other azimuthal projections is attributed to the shape preservation that results from the conformality of the projection.<ref name="German2007"/>