Web Mercator / Pseudo Mercator

New 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.
Classification Cylindrical (non conformant if used with ellipsoid)
Available forms Forward and inverse
Defined area Global
Alias webmerc
Domain 2D
Input type Geodetic coordinates
Output type Projected coordinates

Usage

Example:

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

Parameters

Note

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.

+ellps=<value>

See proj -le for a list of available ellipsoids.

Defaults to “GRS80”.

+lon_0=<value>

Longitude of projection center.

Defaults to 0.0.

+x_0=<value>

False easting.

Defaults to 0.0.

+y_0=<value>

False northing.

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