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| [[File:Winkel triple projection SW.jpg|300px|thumb|Winkel tripel projection of the world. 15° graticule.]]
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| [[File:Tissot indicatrix world map Winkel Tripel proj.svg|thumb|300px|The Winkel tripel projection with [[Tissot's indicatrix]] of deformation]]
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| The '''Winkel tripel projection''' ('''Winkel III'''), a modified azimuthal [[map projection]], is one of three projections proposed by Oswald Winkel in 1921. The projection is the arithmetic mean of the [[equirectangular projection]] and the [[Aitoff projection]]:<REF name="Snyder"/> The name ''Tripel'' (German for "triple") refers to Winkel's goal of minimizing three [[Map projection#Metric properties of maps|kinds of distortion]]: area, direction and distance.<REF name="winkel.org"/>
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| ==Algorithm==
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| :<math>x = \frac{1}{2}\left[\lambda \cos \varphi_1 + \frac{2 \cos \varphi\sin \frac{\lambda}{2}}{\mathrm{sinc}\,\alpha}\right]</math>
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| :<math>y = \frac{1}{2}\left[\varphi + \frac{\sin \varphi}{\mathrm{sinc}\,\alpha}\right]</math>
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| where <math>\lambda</math> is the longitude minus that of the central meridian of the projection, <math>\varphi</math> is the latitude, <math>\varphi_1</math> is the standard parallel for the [[equirectangular projection]], and
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| :<math>\alpha = \arccos\left[\cos\varphi \cos \frac{\lambda}{2} \right]</math>
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| <math>\mathrm{sinc}\,\alpha</math> is the [[sinc function|unnormalized cardinal sine]] function (with the discontinuity removed). In his proposal, Winkel set :
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| :<math>\varphi_1 = \arccos \frac{2}{\pi}\,</math>
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| A [[Closed-form expression|closed-form]] [[Inverse function|inverse mapping]] does not exist, and computing the inverse numerically is somewhat complicated.<ref name="Ipbüker"/>
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| ==Comparison with other projections==
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| Goldberg and [[J. Richard Gott|Gott]] show that the Winkel tripel fares well against several other projections analyzed against their measures of distortion, producing small distance errors, small combinations of [[Tissot's indicatrix|Tissot indicatrix]] ellipticity and area errors, and the smallest [[skewness]] of any of the projections they studied.<ref name="Goldberg-Gott"/>
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| By a different metric, Capek’s “Q”, the Winkel tripel ranked ninth among a hundred map projections of the world, behind the common [[Eckert IV projection]] and [[Robinson projection]]s.<ref name="Capek"/>
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| In 1998, the Winkel tripel projection replaced the [[Robinson projection]] as the standard projection for world maps made by the [[National Geographic Society]]. Many educational institutes and textbooks followed National Geographic's example in adopting the projection, and most of those still use it.<ref>{{cite web|title=NG Maps Print Collection - World Political Map (Bright Colored)|url=http://maps.nationalgeographic.com/maps/print-collection/world-map-bright.html|publisher=National Geographic Society|accessdate=1 October 2013|quote=This latest world map … features the Winkel Tripel projection to reduce the distortion of land masses as they near the poles.}}</ref><ref>{{cite web|title=Selecting a Map Projection - National Geographic Education|url=http://education.nationalgeographic.com/education/media/selecting-map-projection/?ar_a=1|publisher=National Geographic Society|accessdate=1 October 2013}}</ref>
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| ==See also==
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| {{Portal|Atlas}}
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| * [[List of map projections]]
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| == References ==
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| {{Reflist|refs=
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| <ref name="Snyder">{{cite book
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| | title = Flattening the Earth: Two Thousand Years of Map Projections
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| | last = Snyder
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| | first = John P.
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| | authorlink = John P. Snyder
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| | year = 1993
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| | publisher = University of Chicago Press
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| | location = Chicago
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| | isbn = 0-226-76747-7
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| | pages = 231–232
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| | url = http://books.google.com/books?id=0UzjTJ4w9yEC&pg=PA282&dq=winkel
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| | accessdate = 2011-11-14
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| }}</ref>
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| <ref name="winkel.org">{{cite web
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| | url = http://www.winkel.org/other/Winkel%20Tripel%20Projections.htm
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| | title = Winkel Tripel Projections
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| | work = Winkel.org
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| | accessdate = 2011-11-14
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| }}</ref>
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| <ref name="Goldberg-Gott">{{cite journal
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| | url = http://www.physics.drexel.edu/~goldberg/projections/goldberg_gott.pdf
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| | title = Flexion and Skewness in Map Projections of the Earth
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| | year = 2007
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| | first1 = David M.
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| | last1 = Goldberg
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| | first2 = J. Richard
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| | last2 = Gott III
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| | journal = Cartographica
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| | volume = 42
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| | issue = 4
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| | pages = 297–318
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| | accessdate = 2011-11-14
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| }}</ref>
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| <ref name="Capek">{{cite journal
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| | last = Capek
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| | first = Richard
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| | year = 2001
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| | url = http://icaci.org/documents/ICC_proceedings/ICC2001/icc2001/file/f24014.doc
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| | title = Which is the best projection for the world map?
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| | journal = Proceedings of the 20th International Cartographic Conference
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| | location = Beijing, China
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| | volume = 5
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| | pages = 3084–93
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| | accessdate = 2011-11-14
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| }}</ref>
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| <ref name="Ipbüker">{{cite journal | |
| | last = Ipbüker and Bildirici
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| | first = Cengizhan and I.Öztug
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| | year = 2002
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| | url = http://atlas.selcuk.edu.tr/paperdb/papers/130.pdf
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| | title = A GENERAL ALGORITHM FOR THE INVERSE TRANSFORMATION OF MAP PROJECTIONS USING JACOBIAN MATRICES
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| | journal = Proceedings of the Third International Symposium Mathematical & Computational Applications
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| September 4-6, 2002. Konya, Turkey
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| | location = Selcuk, Turkey
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| | pages = 175–182
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| }}</ref>
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| }}
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| == External links ==
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| * [http://www.radicalcartography.net/?projectionref Table of common projections]
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| {{Map Projections}}
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| [[Category:Cartographic projections]]
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