Gnomonic projection
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adding link to ja version and correcting mathematical expressions.
[[Image:Usgs map gnomic.PNG|frame|right|Examples of gnomonic projections]]
The '''gnomonic [[map projection]]''' displays all [[great circle]]s as straight lines.
Thus the shortest route between two locations in reality corresponds to that on the [[map]]. This is achieved by projecting, with respect to the center of the [[Earth]] (hence perpendicular to the surface), the Earth's surface onto a [[tangent]] plane. The least distortion occurs at the tangent point. Less than half of the [[sphere]] can be projected onto a finite map.
Since [[Meridian_%28geography%29|Meridians]] and the [[Equator]] are great circles, they are always shown as straight lines.
*If the tangent point is one of the [[Poles]] then the meridians are radial and equally spaced. The equator is at [[infinity]] in all directions. Other [[parallels]] are depicted as concentric [[circle]]s.
*If the tangent point is on the equator then the meridians are parallel but not equally spaced. The equator is a straight line perpendicular to the meridians. Other parallels are depicted as [[hyperbola]]e.
*In other cases the meridians are radially outward straight lines from a Pole, but not equally spaced. The equator is a straight line that is perpendicular to only one meridian (which again demonstrates that the projection is not [[conformal map|conformal]]).
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[[Image:Gnomonic_Projection_Polar.jpg|thumb|right|Gnomonic projection of Earth centred on the geographic North Pole]]
As for all [[azimuth]]al projections, angles from the tangent point are preserved. The map distance from that point is a function ''r''(''d'') of the true distance ''d'', given by
:<math> r(d) = c \tan (d / R)</math>
where ''R'' is the radius of the Earth. The radial scale is
:<math> r'(d) = \frac{c}{2R \cos^2(d/2R)} </math>
and the [[transverse]] scale
: <math> \frac{c}{2R \cos(d/2R)} </math>
so the transverse scale increases outwardly, and the radial scale even more.
The gnomonic projection is said to be the oldest map projection, developed by [[Thales]] in the [[6th century BC]].
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.
==History==
In [[1946]] [[Buckminster Fuller|Reginald Buckminster Fuller]] patented the Gnomonic Projection in his [[cuboctahedron|cuboctahedral]] version of the [[Dymaxion Map]]. The 1954 [[icosahedron|icosahedral]] version he published under the title of '''AirOcean World Map''', and this is the version most commonly referred to today.
==External links==
*http://www.bfi.org/node/25 Description of the Fuller Projection map from the Buckminster Fuller Institute
*http://erg.usgs.gov/isb/pubs/MapProjections/projections.html#gnomonic Explanations of projections by [[USGS]]
*http://www.3dsoftware.com/Cartography/USGS/MapProjections/Azimuthal/Gnomonic/
*http://exchange.manifold.net/manifold/manuals/6_userman/mfd50Gnomonic.htm
*http://mathworld.wolfram.com/GnomonicProjection.html
*http://members.shaw.ca/quadibloc/maps/maz0201.htm
* [http://www.radicalcartography.net/?projectionref Table of examples and properties of all common projections], from radicalcartography.net
== References ==
{{cite book | author=Snyder, John P. | title=Map Projections - A Working Manual. U.S. Geological Survey Professional Paper 1395 | publisher =United States Government Printing Office, Washington, D.C | year=1987 | id = }} This paper can be downloaded from [http://pubs.er.usgs.gov/pubs/pp/pp1395 USGS pages]
[[Category:Projective geometry]]
[[Category:Navigation]]
[[Category:Cartographic projections]]
[[ca:Projecció azimutal gnomònica]]
[[cs:Gnómonická projekce]]
[[de:Gnomonische Projektion]]
[[es:Proyección gnomónica]]
[[it:Proiezione gnomonica]]
[[ja:心射方位図法]]
[[nl:Gnomonische projectie]]
[[pt:Projeção gnomônica]]