# 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 $\lambda_0$ 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

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