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 +units=m
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.

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.

+lat_0=<value>

Latitude 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: +lat_0=1.5708r

See Projection Units for more information.

+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.

Note

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

Example: +lat_ts=1.5708r

See Projection Units for more information.

+x_0=<value>

False easting.

Defaults to 0.0.

+y_0=<value>

False northing.

Defaults to 0.0.

+ellps=<value>

The name of a built-in ellipsoid definition.

See Ellipsoids for more information, or execute proj -le for a list of built-in ellipsoid names.

Defaults to “GRS80”.

+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.

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