Miller Cylindrical

The Miller cylindrical projection is a modified Mercator projection, proposed by Osborn Maitland Miller in 1942. The latitude is scaled by a factor of \(\frac{4}{5}\), projected according to Mercator, and then the result is multiplied by \(\frac{5}{4}\) to retain scale along the equator.

Classification

Neither conformal nor equal area cylindrical

Available forms

Forward and inverse spherical

Defined area

Global, but best used near the equator

Alias

mill

Domain

2D

Input type

Geodetic coordinates

Output type

Projected coordinates

Miller Cylindrical

proj-string: +proj=mill

Usage

The Miller Cylindrical projection is used for world maps and in several atlases, including the National Atlas of the United States (USGS, 1970, p. 330-331) [Snyder1987].

Example using Central meridian 90°W:

$ echo -100 35 | proj +proj=mill +lon_0=90w
-1113194.91      4061217.24

Parameters

Note

All parameters for the projection are optional.

+lon_0=<value>

Longitude of projection center.

Defaults to 0.0.

Note

The default convention is to interpret this value as decimal degrees. To specify radians instead, follow the value with the “r” character.

Example: +lon_0=1.5708r

See Projection Units for more information.

+R=<value>

Radius of the sphere, given in meters. If used in conjunction with +ellps, +R takes precedence.

See Ellipsoid size parameters for more information.

+x_0=<value>

False easting.

Defaults to 0.0.

+y_0=<value>

False northing.

Defaults to 0.0.

Mathematical definition

The formulas describing the Miller projection are all taken from [Snyder1987].

Forward projection

\[x = \lambda\]
\[y = 1.25 * \ln \left[ \tan \left(\frac{\pi}{4} + 0.4 * \phi \right) \right]\]

Inverse projection

\[\lambda = x\]
\[\phi = 2.5 * ( \arctan \left[ e^{0.8 * y} \right] - \frac{\pi}{4} )\]

Further reading

  1. Wikipedia