Eckert IV projection

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

Forward formulas

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

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