Web Mercator / Pseudo Mercator

Added in version 5.1.0.

The Web Mercator / Pseudo Mercator projection is a cylindrical map projection. This is a variant of the regular Mercator projection, except that the computation is done on a sphere, using the semi-major axis of the ellipsoid.

From Wikipedia:

This projection is widely used by the Web Mercator, Google Web Mercator, Spherical Mercator, WGS 84 Web Mercator[1] or WGS 84/Pseudo-Mercator is a variant of the Mercator projection and is the de facto standard for Web mapping applications. [...] It is used by virtually all major online map providers [...] Its official EPSG identifier is EPSG:3857, although others have been used historically.


Cylindrical (non conformant if used with ellipsoid)

Available forms

Forward and inverse

Defined area






Input type

Geodetic coordinates

Output type

Projected coordinates



$ echo 2 49 | proj +proj=webmerc +datum=WGS84
222638.98       6274861.39



All parameters for the projection are optional, except the ellipsoid definition, which is WGS84 for the typical use case of EPSG:3857. In which case, the other parameters are set to their default 0 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".


Central meridian/longitude of natural origin, longitude of origin or longitude of false origin (naming and meaning depend on the projection method).

Defaults to 0.0.


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=1.570796r

See Projection Units for more information.


False easting, easting at false origin or easting at projection centre (naming and meaning depend on the projection method). Always in meters.

Defaults to 0.0.


False northing, northing at false origin or northing at projection centre (naming and meaning depend on the projection method). Always in meters.

Defaults to 0.0.

Mathematical definition

The formulas describing the Mercator projection are all taken from G. Evenden's libproj manuals [Evenden2005].

Forward projection

\[x = \lambda\]
\[y = \ln \left[ \tan \left(\frac{\pi}{4} + \frac{\phi}{2} \right) \right]\]

Inverse projection

\[\lambda = {x}\]
\[\phi = \frac{\pi}{2} - 2 \arctan \left[ e^{-y} \right]\]

Further reading

  1. Wikipedia