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:

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

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]