Mollweide projection
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2008-06-22T01:15:26Z
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/* Properties */ + map
[[Image:mollweide-projection.jpg|450px|thumb|right| A Mollweide projection of the [[Earth]].]]
The '''Mollweide projection''' is a [[Map projection#Pseudocylindrical|pseudocylindrical map projection]] generally used for global maps of the world (or sky). Also known as the '''Babinet projection''', '''homolographic projection''', or '''elliptical projection'''. As its more explicit moniker ''Mollweide equal area projection'' indicates, it sacrifices fidelity to angle and shape in favor of accurate depiction of 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]] – [[1825]]) of [[Leipzig]] in [[1805]] as an improvement upon the [[Mercator projection]]. It was popularized by Jacques Babinet in [[1857]], giving it the name '''homalographic projection'''<ref>''Flattening the Earth: Two Thousand Years of Map Projections'', John P. Snyder, 1993, pp.112-113, ISBN 0-226-76747-7.</ref>.
The projection is:
:<math>x = \frac{2 \sqrt 2}{\pi} \lambda \cos\left(\theta \right)</math>
:<math>y = \sqrt 2 \sin\left(\theta \right)</math>
where <math>\theta\,</math> is an auxiliary angle defined by
:<math>2 \theta + \sin(2 \theta) = \pi \sin(\phi)\,</math> (1)
and <math>\,\lambda</math> is the longitude from the central meridian, and <math>\,\phi</math> is the latitude.
Equation (1) may be solved with rapid convergence (but slow at the poles) using a Newton-Raphson iteration:
:<math> \delta\theta' = -(\theta' + \sin(\theta') - \pi \sin(\phi))/(1+\cos(\theta'))\,</math>
For the first iteration <math>\,\phi</math> can be used as the first trial <math>\,\theta'</math>
Final <math>\,\theta</math> is calculated as follows:
:<math>\theta = \theta'/2\,</math>
==Properties==
[[Image:Tissot_indicatrix_world_map_Mollweide_proj.svg|thumb|right|350px|The Mollweide projection with [[Tissot's Indicatrix]] of deformation]]
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 corners, although not as severe as in the [[sinusoidal projection]].
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.
The Mollweide, or its properties, has insprired the creation of several other projections, including the [[Goode homolosine projection|Goode's homolosine]], [[van der Grinten projection|van der Grinten]] and the [[Boggs 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.249-252</ref>.
==References==
<references/>
==External links==
* [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]
[[Category:Cartographic projections]]
[[ca:Projecció de Mollweide]]
[[cs:Mollweidovo zobrazení]]
[[de:Mollweide-Projektion]]
[[es:Proyección de Mollweide]]
[[it:Proiezione di Mollweide]]
[[nl:Mollweideprojectie]]
[[ja:モルワイデ図法]]
[[pl:Odwzorowanie Mollweidego]]
[[pt:Projeção de Mollweide]]