Eckert IV projection

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Eckert IV projection of the world.

The Eckert IV projection is an equal-area 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]


Forward formulas[edit]

Given a radius of sphere R, central meridian λ₀ and a point with geographical latitude φ and longitude λ, plane coordinates 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.

θ can be solved for 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]


  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]