# Miller cylindrical projection

A Miller projection of the Earth.

The Miller cylindrical projection is a modified Mercator projection, proposed by Osborn Maitland Miller (1897–1979) in 1942. The latitude is scaled by a factor of 4/5, projected according to Mercator, and then the result is multiplied by 5/4 to retain scale along the equator.[1] Hence:

$x = \lambda$
$y = \frac{5}{4}\ln\left[\tan\left(\frac{\pi}{4} + \frac{2\varphi}{5}\right)\right] = \frac{5}{4}\sinh^{-1}\left(\tan\frac{4\varphi}{5}\right)$

or inversely,

$\lambda = x$
$\varphi = \frac{5}{2}\tan^{-1}\left(e^\frac{4 y}{5}\right) - \frac{5\pi}{8} = \frac{5}{4}\tan^{-1}\left(\sinh\frac{4 y}{5}\right)$

where λ is the longitude from the central meridian of the projection, and $\varphi\,$ is the latitude.[2] Meridians are thus about 0.733 the length of the equator.

In GIS applications, this projection is known as: "EPSG:54003 - World Miller Cylindrical"