Miller cylindrical projection

The Miller cylindrical projection is a modified Mercator projection, proposed by Osborn Maitland Miller 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:

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

A Miller projection of the Earth.

Miller projection with 1,000 km indicatrices of distortion.

or inversely,

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

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

In GIS applications, this projection is known as: "ESRI:54003 - World Miller Cylindrical"[3]

Compact Miller projection is similar to Miller but spacing between parallels stops growing after 55 degrees.[4]