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[[File:Mollweide projection SW.jpg|300px|thumb|Mollweide projection of the world.]]
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[[Image:Tissot indicatrix world map Mollweide proj.svg|thumb|300px|The Mollweide projection with [[Tissot's indicatrix]] of deformation]]
The '''Mollweide projection''' is a [[Map projection|pseudocylindrical map projection]] generally used for global maps of the world (or sky). Also known as the '''Babinet projection''', '''homalographic projection''', '''homolographic projection''', and '''elliptical projection'''. As its more explicit name '''Mollweide equal area projection''' indicates, it sacrifices accuracy of angle and shape in favor of accurate proportions in area. It is used primarily where accurate representation of area takes precedence over shape, for instance small maps depicting global distributions.


The projection was first published by mathematician and astronomer [[Karl Mollweide|Karl (or Carl) Brandan Mollweide]] (1774 &ndash; 1825) of [[Leipzig]] in 1805. It was popularized by [[Jacques Babinet]] in 1857, giving it the name '''homalographic projection'''. The variation '''homolographic''' arose from frequent nineteenth century usage in star atlases.<ref>''Flattening the Earth: Two Thousand Years of Map Projections'', John P. Snyder, 1993, pp.&nbsp;112&ndash;113, ISBN 0-226-76747-7.</ref>
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[[File:Ilc 9yr moll4096.png|thumb|9 year [[WMAP]] image (2012) of the [[Cosmic Microwave Background Radiation]].<ref name="Space-20121221">{{cite web |last=Gannon |first=Megan |title=New 'Baby Picture' of Universe Unveiled |url=http://www.space.com/19027-universe-baby-picture-wmap.html|date=December 21, 2012 |publisher=[[Space.com]] |accessdate=December 21, 2012 }}</ref><ref name="arXiv-20121220">{{cite journal |last=Bennett |first=C.L. |last2=Larson |first2=L.|last3=Weiland |first3=J.L. |last4=Jarosk |first4= N. |last5=Hinshaw |first5=N. |last6=Odegard|first6=N. |last7=Smith |first7=K.M. |last8=Hill |first8=R.S. |last9=Gold |first9=B.|last10=Halpern |first10=M. |last11=Komatsu |first11=E. |last12=Nolta |first12=M.R.|last13=Page |first13=L. |last114=Spergel |first14=D.N. |last15=Wollack |first15=E.|last16=Dunkley |first16=J. |last17=Kogut |first17=A. |last18=Limon |first18=M. |last19=Meyer|first19=S.S. |last20=Tucker |first20=G.S. |last21=Wright |first21=E.L. |title=Nine-Year Wilkinson Microwave Anisotropy Probe (WMAP) Observations: Final Maps and Results|url=http://arxiv.org/abs/1212.5225 |arxiv=1212.5225 |date=December 20, 2012|accessdate=December 22, 2012 }}</ref> Projected using the Mollweide projection.]]
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[[Image:GLODAP sea-surf CFC12 AYool.png|thumb|Sea-surface [[freon]] levels measured by the [[Global Ocean Data Analysis Project]]. Projected using the Mollweide projection.]]
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The Mollweide is a [[Map projection#Pseudocylindrical|pseudocylindrical]] projection in which the [[equator]] is represented as a straight horizontal line perpendicular to a central [[meridian (geography)|meridian]] one-half its length. The other [[parallel (latitude)|parallel]]s compress near the poles, while the other meridians are equally spaced at the equator. The meridians at 90 degrees east and west form a perfect circle, and the whole earth is depicted in a proportional 2:1 ellipse. The proportion of the area of the ellipse between any given parallel and the equator is the same as the proportion of the area on the globe between that parallel and the equator, but at the expense of shape distortion, which is significant at the perimeter of the ellipse, although not as severe as in the [[sinusoidal projection]].
 
 
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Shape distortion may be diminished by using an ''interrupted'' version. A ''sinusoidal interrupted'' Mollweide projection discards the central meridian in favor of alternating half-meridians which terminate at right angles to the equator. This has the effect of dividing the globe into lobes  shape. In contrast, a ''parallel interrupted'' Mollweide projection uses multiple disjoint central meridians, giving the effect of multiple ellipses joined at the equator. More rarely, the project can be drawn obliquely to shift the areas of distortion to the oceans, allowing the continents to remain truer to form.
 
 
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The Mollweide, or its properties, has inspired the creation of several other projections, including the [[Goode homolosine projection|Goode's homolosine]], [[van der Grinten projection|van der Grinten]] and the [[Boggs eumorphic projection|Boggs eumorphic]].<ref>[http://pubs.er.usgs.gov/usgspubs/pp/pp1395 ''Map Projections – A Working Manual''], [[United States Geological Survey|USGS]] Professional Paper 1395, John P. Snyder, 1987, pp.&nbsp;249&ndash;252</ref>
 
 
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==Mathematical Formulation==
 
The projection transforms from latitude and longitude to map coordinates through the following equations:<ref name="MathWorldMollweide">{{MathWorld|title=Mollweide Projection|urlname=MollweideProjection|author=Weisstein, Eric W.}}</ref>
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:<math>x = \frac{2 \sqrt 2}{\pi} \left( \lambda - \lambda_{0} \right) \cos \left( \theta \right),</math>
</ul>
 
:<math>y = \sqrt 2 \sin \left( \theta \right),\,</math>
 
where <math>\theta\,</math> is an auxiliary angle defined by
 
:<math>2 \theta + \sin \left( 2 \theta \right) = \pi \sin \left( \varphi \right) \qquad (1)</math>
 
and ''λ'' is the longitude, ''λ<sub>0</sub>'' is the central meridian, and ''φ'' is the latitude.
 
Equation (1) may be solved with rapid convergence (but slow near the poles) using [[Newton's method|Newton&ndash;Raphson]] iteration:<ref name="MathWorldMollweide"/>
 
:<math> \theta_0 = \varphi,\,</math>
 
:<math> \theta_{n+1} = \theta_n - \frac{2 \theta_n + \sin \left( 2 \theta_n \right) - \pi \sin \left( \varphi \right)}{2 + 2 \cos \left( 2 \theta_n \right)}.\,</math>
 
If ''φ'' = ±π/2, then also θ = ±π/2. In that case the iteration should be bypassed; otherwise, [[division by zero]] may result.
 
There exists a [[closed-form expression|closed-form]] inverse transformation:<ref name="MathWorldMollweide"/>
 
:<math> \varphi = \sin^{-1} \left[ \frac{2 \theta + \sin \left( 2 \theta \right)}{\pi} \right], \,</math>
:<math> \lambda = \lambda_{0} + \frac{\pi x}{2 \sqrt{2} \cos \theta}, \,</math>
 
where ''θ'' can be found by the relation
 
:<math>\theta = \sin^{-1} \left( \frac{y}{\sqrt{2}} \right). \,</math>
 
The inverse transformations allow one to find the latitude and longitude corresponding to the map coordinates ''x'' and ''y''.
 
==See also==
{{Portal|Atlas}}
* [[List of map projections]]
* [[Aitoff projection]]
* [[Hammer projection]]
 
==References==
{{reflist}}
 
==External links==
{{Commonscat|Mollweide projection}}
*[http://www.uff.br/mapprojections/Mollweide_en.html An interactive JAVA applet to study deformations (area, distance and angle) of the Mollweide Map Projection]
*[http://mathworld.wolfram.com/MollweideProjection.html Mollweide Projection at Mathworld]
 
{{Map Projections}}
 
[[Category:Cartographic projections]]
[[Category:Equal-area projections]]

Latest revision as of 07:35, 13 December 2014

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