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, spherical projection
Defined area	Global, but best used near the equator
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.

+R=<value>¶: Radius of the sphere given in meters. If used in conjunction with +ellps +R takes precedence.

+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¶

\[\lambda = {x}\]

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