en>Mogism: /* Beam characteristics: curvature radius, width, and polarization */Cleanup/Typo fixing, typos fixed: ,, → , using AWB

2013-07-30T18:59:24Z

Beam characteristics: curvature radius, width, and polarization: Cleanup/Typo fixing, typos fixed: ,, → , using AWB

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[[File:Ecker IV projection SW.jpg|450px|thumb|Eckert IV projection of the world.]]

The '''Eckert IV projection''' is a [[Map projection|pseudocylindrical map projection]]. The length of polar line is half that of the equator, and lines of longitude are semiellipses, or portions of [[ellipse]]s. It was first described by Max Eckert in 1906.<ref name="Snyder89" />
==Formulas==
===Forward formulas===
Given a radius of sphere <math>R</math>, central meridian <math>\lambda_0</math> and a point with polar coordinates <math>(\varphi,\lambda)</math>, <math>x</math> and <math>y</math> can be computed using the following formulas:
: <math>x = \frac{2}{\sqrt{4\pi + \pi^2}} R\, (\lambda - \lambda_0)(1 + \cos \theta) \approx 0.4222382\, R\, (\lambda - \lambda_0)(1 + \cos \theta)</math>,
: <math>y = 2 \sqrt{\frac{\pi}{4 + \pi}} R \sin \theta \approx 1.3265004\, R \sin \theta</math>,
: where <math>\theta + \sin \theta \cos \theta + 2 \sin \theta = \left(2 + \frac \pi 2\right) \sin \varphi</math>. This equation can be solved numerically using [[Newton's method]].<ref name="Snyder87" />

===Inverse formulas===
: <math>\theta = \arcsin \left[y \frac{\sqrt{4 + \pi}}{2 \sqrt\pi R}\right] \approx \arcsin \left[\frac{y}{1.3265004\, R}\right]</math>
: <math>\varphi = \arcsin \left[\frac{\theta + \sin \theta \cos \theta + 2 \sin \theta}{2 + \frac \pi 2}\right]</math>
: <math>\lambda = \lambda_0 + x \frac{\sqrt{4\pi + \pi^2}}{2R (1 + \cos \theta)} \approx \lambda_0 + \frac{x}{0.4222382\, R\, (1 + \cos \theta)}</math>

==See also==
* [[List of map projections]]
*[[Eckert II projection]]
*[[Eckert VI projection]]
*[[Max Eckert-Greifendorff]]

== References ==
{{Reflist|refs=
<ref name="Snyder89">{{cite book
| title = An Album of Map Projections
| last = Snyder
| first = John P.
| authorlink = John P. Snyder
| year = 1989
| publisher = [[United States Geological Survey|USGS]]
| location = Denver
| series = Professional Paper 1453
| page = 60
| url = http://pubs.er.usgs.gov/publication/pp1453
}}</ref>
<ref name="Snyder87">{{cite book
| title = Map Projections – A Working Manual
| last = Snyder
| first = John P.
| authorlink = John P. Snyder
| year = 1987
| publisher = [[United States Geological Survey|USGS]]
| location = Denver
| series = Professional Paper 1395
| isbn = 0-226-76747-7
| pages = 253–258
| url = http://pubs.er.usgs.gov/usgspubs/pp/pp1395
| accessdate = 2013-07-24
}}</ref>
}}

==External links==
*[http://mathworld.wolfram.com/EckertIVProjection.html Eckert IV projection at Mathworld]
{{Map Projections}}

[[Category:Cartographic projections]]
[[Category:Equal-area projections]]

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en>Mogism: /* Beam characteristics: curvature radius, width, and polarization */Cleanup/Typo fixing, typos fixed: ,, → , using AWB