View source for Gnomonic projection

{{Short description|Projection of a sphere through its center onto a plane}}
{{No footnotes|date=February 2019}}
[[File:Gnomonic projection SW.jpg|300px|thumb|Gnomonic projection of a portion of the north hemisphere centered on the geographic North Pole]]
[[File:Gnomonic with Tissot's Indicatrices of Distortion.svg|thumb|The gnomonic projection with [[Tissot's indicatrix]] of deformation]]

A '''gnomonic projection''', also known as a '''central projection''' or '''rectilinear projection''', is a [[perspective projection]] of a [[sphere]], with center of projection at the sphere's [[center (geometry)|center]], onto any [[plane (geometry)|plane]] not passing through the center, most commonly a [[tangent]] plane. Under gnomonic projection every [[great circle]] on the sphere is projected to a [[straight line]] in the plane (a great circle is a [[geodesic]] on the sphere, the shortest path between any two points, analogous to a straight line on the plane).<ref>{{cite journal |last1=Williams |first1=C.E. |last2=Ridd |first2=M.K. |year=1960 |title=Great Circles and the Gnomonic Projection |journal=The Professional Geographer |volume=12 |number=5 |pages=14–16 |doi=10.1111/j.0033-0124.1960.125_14.x }}</ref> More generally, a gnomonic projection can be taken of any [[n-sphere|{{mvar|n}}-dimensional hypersphere]] onto a [[hyperplane]].

The projection is the {{mvar|n}}-dimensional generalization of the [[tangent (trigonometry)|trigonometric tangent]] which maps from the [[circle]] to a straight line, and as with the tangent, every pair of [[antipodal point]]s on the sphere projects to a single point in the plane, while the points on the plane through the sphere's center and parallel to the image plane project to [[point at infinity|points at infinity]]; often the projection is considered as a [[Bijection|one-to-one correspondence]] between points in the hemisphere and points in the plane, in which case any finite part of the image plane represents a portion of the hemisphere.<ref name=Snyder>{{cite book |last=Snyder |first=John P. |title=Map Projections – A Working Manual |series=U.S. Geological Survey Professional Paper |volume=1395 |publisher=United States Government Printing Office |year=1987 |pages=164–168 |location=Washington, D.C. |url=https://archive.org/details/Snyder1987MapProjectionsAWorkingManual/page/n178 |doi=10.3133/pp1395 }}</ref>

The gnomonic projection is [[azimuthal projection|azimuthal]] (radially symmetric). No shape distortion occurs at the center of the projected image, but distortion increases rapidly away from it.

The gnomonic projection originated in [[astronomy]] for constructing [[sundial]]s and charting the [[celestial sphere]]. It is commonly used as a geographic [[map projection]], and can be convenient in [[navigation]] because great-circle courses are plotted as straight lines. [[Rectilinear lens|Rectilinear photographic lens]]es make a perspective projection of the world onto an image plane; this can be thought of as a gnomonic projection of the [[image sphere]] (an abstract sphere indicating the direction of each ray passing through a [[pinhole camera model|camera modeled as a pinhole]]). The gnomonic projection is used in [[crystallography]] for analyzing the orientations of lines and planes of crystal structures. It is used in [[structural geology]] for analyzing the orientations of fault planes. In [[computer graphics]] and computer representation of spherical data, [[cube mapping]] is the gnomonic projection of the image sphere onto six faces of a [[cube]].

In mathematics, the space of [[orientation (geometry)|orientations]] of undirected lines in [[3-dimensional space]] is called the [[real projective plane]], and is typically pictured either by the "projective sphere" or by its gnomonic projection. When the [[angle]] between lines is imposed as a [[distance function|measure of distance]], this space is called the [[elliptic plane]]. The gnomonic projection of the 3-sphere of [[unit quaternions]], points of which represent 3-dimensional rotations, results in [[Rodrigues vector]]s. The gnomonic projection of the [[hyperboloid of two sheets]], treated as a model for the [[hyperbolic plane]], is called the [[Beltrami–Klein model]].

==History==
The gnomonic [[Map projection|projection]] is said to be the oldest map projection, speculatively attributed to [[Thales]] who may have used it for star maps in the 6th century BC.{{r|Snyder}} The path of the shadow-tip or light-spot in a [[sundial#Nodus-based sundials|nodus-based sundial]] traces out the same [[hyperbola]]e formed by parallels on a gnomonic map.

==Properties==
The gnomonic projection is from the centre of a sphere to a plane tangent to the sphere (Fig 1 below). The sphere and the plane touch at the tangent point. Great circles transform to straight lines via the gnomonic projection. Since [[Meridian (geography)|meridians]] (lines of longitude) and the [[equator]] are great circles, they are always shown as straight lines on a gnomonic map. Since the projection is from the centre of the sphere, a gnomonic map can represent less than half of the area of the sphere. Distortion of the scale of the map increases from the centre (tangent point) to the periphery.<ref name=Snyder/>

*If the tangent point is one of the [[Geographical pole|poles]] then the meridians  are radial and equally spaced (Fig 2 below). The equator cannot be shown as it is at [[infinity]] in all directions. Other [[Circle of latitude|parallels]] (lines of latitude) are depicted as concentric [[circle]]s.

*If the tangent point is on the equator then the meridians are parallel but not equally spaced (Fig 3 below). The equator is a straight line perpendicular to the meridians. Other parallels are depicted as [[hyperbola]]e.

*If the tangent point is not on a pole or the equator, then the meridians are radially outward straight lines from a pole, but not equally spaced (Fig 4 below). The equator is a straight line that is perpendicular to only one meridian, indicating that the projection is not [[conformal map|conformal]]. Other parallels are depicted as [[conic section]]s.

{{Gallery
|width=200
|align=left
|footer=Figs 2 - 4 are from Snyder (1987) Figure 34{{r|Snyder|p=166}}.
|File:Gnomonic.png 
|Fig 1. A great circle projects to a straight line in the gnomonic projection
|File:Snyder Figure 34 Gnomonic A.jpg
|Fig 2. Gnomonic projection centered on the north pole
|File:Snyder Figure 34 Gnomonic B.jpg
|Fig 3. Gnomonic projection centered on the equator
|File:Snyder Figure 34 Gnomonic C.jpg
|Fig 4. Gnomonic projection centered on latitude 40° north
}}
{{clear}}


As with all [[azimuth]]al projections, angles from the tangent point are preserved. The map distance from that point is a function ''r''(''d'') of the [[great-circle distance|true distance]] ''d'', given by

:<math> r(d) = R\,\tan \frac d R</math>

where ''R'' is the radius of the Earth. The radial scale is

:<math> r'(d) = \frac{1}{\cos^2\frac d R} </math>

and the [[:wikt:transverse|transverse]] scale

: <math> \frac{1}{\cos\frac d R} </math>

so the transverse scale increases outwardly, and the radial scale even more.

==Use==
[[File:Admiralty Chart No 132 Gnomonic Chart of Indian and Southern Oceans, Published 1914.jpg|thumb|Admiralty Gnomonic Chart of the Indian and Southern Oceans, for use in plotting great circle tracks]]
Gnomonic projections are used in [[seismic]] work because seismic waves tend to travel along great circles. They are also used by [[navy|navies]] in plotting [[direction finding]] bearings, since [[radio]] signals travel along great circles. [[Meteor]]s also travel along great circles, with the Gnomonic [[Atlas Brno 2000.0]] being the [[International Meteor Organization|IMO]]'s recommended set of star charts for visual meteor observations. Aircraft and ship navigators use the projection to find the shortest route between start and destination. The track is first drawn on the gnomonic chart, then transferred to a Mercator chart for navigation.

The gnomonic projection is used extensively in [[photography]], where it is called ''[[Rectilinear lens|rectilinear]] projection'', as it naturally arises from the [[pinhole camera model]] where the screen is a plane.<ref>{{cite book | url=https://books.google.com/books?id=GlOzDQAAQBAJ&pg=PA669 | title=Handbook of Digital Image Synthesis: Scientific Foundations of Rendering | isbn=978-1-315-39521-0 | last1=Pegoraro | first1=Vincent | date=12 December 2016 | publisher=CRC Press }}</ref> Because they are equivalent, the same viewer used for photographic panoramas can be used to render gnomonic maps {{PanoLink|Equilateral_with_tissot.jpg}}.

The gnomonic projection is used in astronomy where the tangent point is centered on the object of interest.  The sphere being projected in this case is the celestial sphere, ''R''&nbsp;=&nbsp;1,  and not the surface of the Earth.

In astronomy, gnomic projection star charts of the [[celestial sphere]] can be used by observers to accurately plot the straight line path of a [[meteor]] trail.<ref name=Taibi_2016>{{citation | postscript=.
 | title=Charles Olivier and the Rise of Meteor Science
 | first=Richard | last=Taibi | date=November 25, 2016
 | page=67 | isbn=9783319445182
 | publisher=Springer International Publishing
 | url=https://books.google.com/books?id=nVyXDQAAQBAJ&pg=PA67 }}</ref>

{{comparison_azimuthal_projections.svg}}

==See also==
* [[Local tangent plane]]
* [[List of map projections]]
* [[Beltrami–Klein model]], the analogous mapping of the [[hyperbolic geometry|hyperbolic plane]]

==References==
{{reflist}}

== Further reading ==
* {{cite book |last1=Amorós |first1=José Luis |last2=Buerger |first2=Martin J. |last3=Canut de Amorós |first3=Marisa |year=1975 |chapter=3. The gnomonic projection |title=The Laue Method |pages=55–81 |publisher=Academic Press |doi=10.1016/B978-0-12-057450-6.50006-5 }}
* {{cite journal |last1=Calabretta |first1=Mark R. |last2=Greisen |first2=Eric W. |title=Representations of celestial coordinates in FITS (Paper II) |year=2002 |journal=Astronomy & Astrophysics |volume=395 |pages=1077–1122 |arxiv=astro-ph/0207413 |doi=10.1051/0004-6361:20021327 }}
* {{cite book |last=Boeke |first=Hendrik Enno |author-link=Hendrik Enno Boeke |year=1913 |title=Die gnomonische Projektion in ihrer Anwendung auf kristallographische Aufgaben |language=de |trans-title=The gnomonic projection in its application to crystallographic tasks |publisher=Gebrüder Borntraeger }}
* {{cite journal |last=Bradley |first=A.D. |year=1940 |title=The Gnomonic Projection of the Sphere |journal=American Mathematical Monthly |volume=47 |number=10 |pages=694–699 |doi=10.2307/2302492 |jstor=2302492 }}
* {{cite book |last=De Morgan |first=Augustus |author-link= Augustus De Morgan |year=1836 |title=An explanation of the Gnomonic Projection of the Sphere |publisher=Baldwin & Cradock |url=https://archive.org/details/explanationofgno00demorich }}
* {{cite journal |last=Günther |first=S. |year=1883 |title=Die gnomonische Kartenprojektion |journal=Zeitschrift der Gesellschaft für Erdkunde zu Berlin |volume=18 |pages=137–149 |url=https://books.google.com/books?id=q_E6AQAAIAAJ&pg=PA137 |language=de |trans-title=The gnomonic map projection }}
* {{cite journal |last=Herbert Smith |first=G. F. |year=1903 |title=On the Advantages of the Gnomonic Projection and Its Use in the Drawing of Crystals |journal=Mineralogical Magazine |volume=13 |number=62 |pages=309–322 |doi=10.1180/minmag.1903.013.62.02 |url=https://rruff.info/doclib/MinMag/Volume_13/13-62-309.pdf }}
* {{cite journal |last=Hilton |first=Harold |year=1904 |title=The Gnomonic Net |journal=Mineralogical Magazine |volume=14 |number=63 |pages=18–20 |doi=10.1180/minmag.1904.014.63.05 |url=https://rruff.info/doclib/MinMag/Volume_14/14-63-18.pdf }}
* {{cite journal |last=Palache |first=Charles |year=1920 |title=The gnomonic projection |journal=American Mineralogist |volume=5 |number=4 |pages=67–80 |url=http://www.minsocam.org/ammin/AM5/AM5_67.pdf }}
* {{cite journal |last=Pye |first=Norman |year=1950 |title=The Oblique Gnomonic Projection |journal=Empire Survey Review |volume=10 |number=75 |pages=227–232 |doi=10.1179/sre.1950.10.75.227 }}
* {{cite journal |last=Rogers |first=Austin F. |year=1907 |title=The Gnomonic Projection from a Graphical Standpoint |journal=School of Mines Quarterly |volume=29 |number=1 |pages=24–33 |url=https://archive.org/details/schoolminesquar17chemgoog/page/n53/ }}

==External links==
{{Commons category}}
*[http://mathworld.wolfram.com/GnomonicProjection.html Gnomonic Projection]

{{Map projections}}

[[Category:Map projections]]
[[Category:Navigation]]
[[Category:Projective geometry]]