Equidistant Cylindrical (Plate Carrée)

The simplest of all projections. Standard parallels (0° when omitted) may be specified that determine latitude of true scale (k=h=1).

Classification Conformal cylindrical
Available forms Forward and inverse
Defined area Global, but best used near the equator
Alias eqc
Domain 2D
Input type Geodetic coordinates
Output type Projected coordinates
Equidistant Cylindrical (Plate Carrée)

proj-string: +proj=eqc

Usage

Because of the distortions introduced by this projection, it has little use in navigation or cadastral mapping and finds its main use in thematic mapping. In particular, the plate carrée has become a standard for global raster datasets, such as Celestia and NASA World Wind, because of the particularly simple relationship between the position of an image pixel on the map and its corresponding geographic location on Earth.

The following table gives special cases of the cylindrical equidistant projection.

Projection Name (lat ts=) \(\phi_0\)
Plain/Plane Chart
Simple Cylindrical
Plate Carrée
Ronald Miller—minimum overall scale distortion 37°30′
E.Grafarend and A.Niermann 42°
Ronald Miller—minimum continental scale distortion 43°30′
Gall Isographic 45°
Ronald Miller Equirectangular 50°30′
E.Grafarend and A.Niermann minimum linear distortion 61°7′

Example using EPSG 32662 (WGS84 Plate Carrée):

$ echo 2 47 | proj +proj=eqc +lat_ts=0 +lat_0=0 +lon_0=0 +x_0=0 +y_0=0 +ellps=WGS84 +datum=WGS84 +units=m +no_defs
222638.98       5232016.07

Example using Plate Carrée projection with true scale at latitude 30° and central meridian 90°W:

$ echo -88 30 | proj +proj=eqc +lat_ts=30 +lon_0=90w
192811.01       3339584.72

Parameters

+lon_0=<value>

Longitude of projection center.

Defaults to 0.0.

+lat_0=<value>

Latitude of projection center.

Defaults to 0.0.

+lat_ts=<value>

Latitude of true scale. Defines the latitude where scale is not distorted. Takes precedence over +k_0 if both options are used together.

Defaults to 0.0.

+x_0=<value>

False easting.

Defaults to 0.0.

+y_0=<value>

False northing.

Defaults to 0.0.

+ellps=<value>

See proj -le for a list of available ellipsoids.

Defaults to “GRS80”.

+R=<value>

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

Mathematical definition

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

\(\phi_{ts}\) is the latitude of true scale, that mean the standard parallels where the scale of the projection is true. It can be set with +lat_ts.

\(\phi_0\) is the latitude of origin that match the center of the map. It can be set with +lat_0.

Forward projection

\[x = \lambda \cos \phi_{ts}\]
\[y = \phi - \phi_0\]

Inverse projection

\[\lambda = x / cos \phi_{ts}\]
\[\phi = y + \phi_0\]

Further reading

  1. Wikipedia
  2. Wolfram Mathworld