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:

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

or inversely,

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

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]

References

^ Flattening the Earth: Two Thousand Years of Map Projections, John P. Snyder, 1993, pp. 179, 183, ISBN 0-226-76747-7.
^ "Miller Cylindrical Projection". Wolfram MathWorld. Retrieved 25 March 2015.
^ "Projected coordinate systems". ArcGIS Resources: ArcGIS Rest API. ESRI. Retrieved 16 June 2017.
^ Patterson, Tom; Šavrič, Bojan; Jenny, Bernhard (2015). "Introducing the Patterson Cylindrical Projection". Cartographic Perspectives (78): 77–81. doi:10.14714/CP78.1270.