Eckert IV projection

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Eckert IV projection of the world.

The Eckert IV projection is a pseudocylindrical map projection. The length of polar line is half that of the equator, and lines of longitude are semiellipses, or portions of ellipses. It was first described by Max Eckert in 1906.[1]

Formulas[edit]

Forward formulas[edit]

Given a radius of sphere R, central meridian \lambda_0 and a point with polar coordinates (\varphi,\lambda), x and y can be computed using the following formulas:

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),
y = 2 \sqrt{\frac{\pi}{4 + \pi}} R \sin \theta \approx 1.3265004\, R \sin \theta,
where \theta + \sin \theta \cos \theta + 2 \sin \theta = \left(2 + \frac \pi 2\right) \sin \varphi. This equation can be solved numerically using Newton's method.[2]

Inverse formulas[edit]

\theta = \arcsin \left[y \frac{\sqrt{4 + \pi}}{2 \sqrt\pi R}\right] \approx \arcsin \left[\frac{y}{1.3265004\, R}\right]
\varphi = \arcsin \left[\frac{\theta + \sin \theta \cos \theta + 2 \sin \theta}{2 + \frac \pi 2}\right]
\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)}

See also[edit]

References[edit]

  1. ^ Snyder, John P. (1989). An Album of Map Projections. Professional Paper 1453. Denver: USGS. p. 60. 
  2. ^ Snyder, John P. (1987). Map Projections – A Working Manual. Professional Paper 1395. Denver: USGS. pp. 253–258. ISBN 0-226-76747-7. Retrieved 2013-07-24. 

External links[edit]