en>Yobot: Tagging, removed underlinked tag using AWB (10471)

2014-09-22T15:56:27Z

Tagging, removed underlinked tag using AWB (10471)

en>Wizardman: clean up using AWB

2014-01-19T05:39:26Z

clean up using AWB

← Older revision		Revision as of 07:39, 19 January 2014
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	~~Hello~~, my name ~~is Felicidad but I don~~'~~t like when individuals use my complete title~~. ~~One~~ of the ~~issues I adore most~~ is ~~greeting card gathering but I don~~'~~t have~~ the ~~time lately~~. ~~Interviewing is how I make~~ a ~~living~~ and it's ~~some thing I truly appreciate~~. ~~Delaware is our beginning location~~.<br><br>~~Also visit my~~ web-~~site - [~~http://89.1.13.~~189~~/~~index~~.~~php~~?~~mod~~=~~users~~&~~action~~=~~view~~&id=~~63366 89~~.1.13.~~189~~]		[[File:Winkel triple projection SW.jpg\|300px\|thumb\|Winkel tripel projection of the world. 15° graticule.]]
			[[File:Tissot indicatrix world map Winkel Tripel proj.svg\|thumb\|300px\|The Winkel tripel projection with [[Tissot's indicatrix]] of deformation]]

			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"/>

			==Algorithm==
			:<math>x = \frac{1}{2}\left[\lambda \cos \varphi_1 + \frac{2 \cos \varphi\sin \frac{\lambda}{2}}{\mathrm{sinc}\,\alpha}\right]</math>

			:<math>y = \frac{1}{2}\left[\varphi + \frac{\sin \varphi}{\mathrm{sinc}\,\alpha}\right]</math>

			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

			:<math>\alpha = \arccos\left[\cos\varphi \cos \frac{\lambda}{2} \right]</math>

			<math>\mathrm{sinc}\,\alpha</math> is the [[sinc function\|unnormalized cardinal sine]] function (with the discontinuity removed). In his proposal, Winkel set :

			:<math>\varphi_1 = \arccos \frac{2}{\pi}\,</math>

			A [[Closed-form expression\|closed-form]] [[Inverse function\|inverse mapping]] does not exist, and computing the inverse numerically is somewhat complicated.<ref name="Ipbüker"/>

			==Comparison with other projections==
			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"/>
			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"/>

			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>

			==See also==
			{{Portal\|Atlas}}
			* [[List of map projections]]

			== References ==
			{{Reflist\|refs=
			<ref name="Snyder">{{cite book
			\| title = Flattening the Earth: Two Thousand Years of Map Projections
			\| last = Snyder
			\| first = John P.
			\| authorlink = John P. Snyder
			\| year = 1993
			\| publisher = University of Chicago Press
			\| location = Chicago
			\| isbn = 0-226-76747-7
			\| pages = 231–232
			\| url = http://books.google.com/books?id=0UzjTJ4w9yEC&pg=PA282&dq=winkel
			\| accessdate = 2011-11-14
			}}</ref>
			<ref name="winkel.org">{{cite web
			\| url = http://www.winkel.org/other/Winkel%20Tripel%20Projections.htm
			\| title = Winkel Tripel Projections
			\| work = Winkel.org
			\| accessdate = 2011-11-14
			}}</ref>
			<ref name="Goldberg-Gott">{{cite journal
			\| url = http://www.physics.drexel.edu/~goldberg/projections/goldberg_gott.pdf
			\| title = Flexion and Skewness in Map Projections of the Earth
			\| year = 2007
			\| first1 = David M.
			\| last1 = Goldberg
			\| first2 = J. Richard
			\| last2 = Gott III
			\| journal = Cartographica
			\| volume = 42
			\| issue = 4
			\| pages = 297–318
			\| accessdate = 2011-11-14
			}}</ref>
			<ref name="Capek">{{cite journal
			\| last = Capek
			\| first = Richard
			\| year = 2001
			\| url = http://icaci.org/documents/ICC_proceedings/ICC2001/icc2001/file/f24014.doc
			\| title = Which is the best projection for the world map?
			\| journal = Proceedings of the 20th International Cartographic Conference
			\| location = Beijing, China
			\| volume = 5
			\| pages = 3084–93
			\| accessdate = 2011-11-14
			}}</ref>
			<ref name="Ipbüker">{{cite journal
			\| last = Ipbüker and Bildirici
			\| first = Cengizhan and I.Öztug
			\| year = 2002
			\| url = http://atlas.selcuk.edu.tr/paperdb/papers/130.pdf
			\| title = A GENERAL ALGORITHM FOR THE INVERSE TRANSFORMATION OF MAP PROJECTIONS USING JACOBIAN MATRICES
			\| journal = Proceedings of the Third International Symposium Mathematical & Computational Applications
			September 4-6, 2002. Konya, Turkey
			\| location = Selcuk, Turkey
			\| pages = 175–182
			}}</ref>
			}}


			== External links ==
			* [http://www.radicalcartography.net/?projectionref Table of common projections]

			{{Map Projections}}

			[[Category:Cartographic projections]]

en>ChrisGualtieri: /* Definition */Typo fixing, typos fixed: , → , using AWB

2012-07-21T01:15:24Z

Definition: Typo fixing, typos fixed: , → , using AWB

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Hello, my name is Felicidad but I don't like when individuals use my complete title. One of the issues I adore most is greeting card gathering but I don't have the time lately. Interviewing is how I make a living and it's some thing I truly appreciate. Delaware is our beginning location.<br><br>Also visit my web-site - [http://89.1.13.189/index.php?mod=users&action=view&id=63366 89.1.13.189]

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en>Yobot: Tagging, removed underlinked tag using AWB (10471)

en>Wizardman: clean up using AWB

en>ChrisGualtieri: /* Definition */Typo fixing, typos fixed: , → , using AWB