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| [[Image:Gnomonic.png|thumb|right|300px|Great circles transform to straight lines via gnomonic projection]]
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| [[Image:Usgs map gnomic.PNG|frame|right|Examples of gnomonic projections]]
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| A '''gnomonic [[map projection]]''' displays all [[great circle]]s as straight lines, resulting in any line segment on a gnomonic map showing the shortest route between the segment's two endpoints. This is achieved by casting surface points of the sphere onto a [[tangent]] plane, each landing where a ray from the center of the earth passes through the point on the surface and then on to the plane. No distortion occurs at the tangent point, but distortion increases rapidly away from it. Less than half of the [[sphere]] can be projected onto a finite map. Consequently a rectilinear photographic lens cannot image more than 180 degrees.
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| ==History==
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| The gnomonic projection is said to be the oldest map projection, developed by [[Thales]] in the 6th century BC.
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| 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. | |
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| ==Properties==
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| Since [[Meridian (geography)|meridians]] and the [[equator]] are great circles, they are always shown as straight lines on a gnomonic map.
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| *If the tangent point is one of the [[Geographical pole|poles]] then the meridians are radial and equally spaced. The equator is at [[infinity]] in all directions. Other [[Circle of latitude|parallels]] are depicted as concentric [[circle]]s.
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| *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. The equator is a straight line that is perpendicular to only one meridian, indicating that the projection is not [[conformal map|conformal]].
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| *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.
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| [[File:Gnomonic projection SW.jpg|300px|thumb|Gnomonic projection of the world centered on the geographic North Pole]]
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| 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 true distance ''d'', given by
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| :<math> r(d) = R\,\tan (d/R)</math>
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| where ''R'' is the radius of the Earth. The radial scale is | |
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| :<math> r'(d) = \frac{1}{\cos^2(d/R)} </math>
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| and the [[:wikt:transverse|transverse]] scale
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| : <math> \frac{1}{\cos(d/R)} </math>
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| so the transverse scale increases outwardly, and the radial scale even more.
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| ==Use==
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| 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 pilots use the projection to find the shortest route between start and destination.
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| The gnomonic projection is used extensively in [[photography]], where it is called ''[[Rectilinear_lens|rectilinear]] projection''.
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| ==See also==
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| {{Portal|Atlas}}
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| * [[List of map projections]]
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| *[[Beltrami–Klein model]], the analogous mapping of the [[hyperbolic geometry|hyperbolic plane]]
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| ==References==
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| {{reflist}}
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| *{{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]
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| ==External links==
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| *http://www.bfi.org/node/25 Description of the Fuller Projection map from the Buckminster Fuller Institute
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| *http://erg.usgs.gov/isb/pubs/MapProjections/projections.html#gnomonic Explanations of projections by [[USGS]]
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| *http://exchange.manifold.net/manifold/manuals/6_userman/mfd50Gnomonic.htm
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| *[http://mathworld.wolfram.com/GnomonicProjection.html Gnomonic Projection]
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| *[http://members.shaw.ca/quadibloc/maps/maz0201.htm The Gnomonic Projection]
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| * [http://www.radicalcartography.net/?projectionref Table of examples and properties of all common projections], from radicalcartography.net
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| * [http://sourceforge.net/projects/sphaerica Sphaerica geometry software], is capable of displaying spherical geometry constructions in gnomonic projection
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| {{Map Projections}}
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| [[Category:Cartographic projections]]
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| [[Category:Navigation]]
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| [[Category:Projective geometry]]
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Oscar is what my spouse enjoys to contact me and I completely dig that name. Hiring is his occupation. The preferred pastime for my children and me is to perform baseball but I haven't produced a dime with it. Puerto Rico is where he's been living for many years and he will never move.
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