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Web Mercator / Pseudo Mercator

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New in version 5.1.0.

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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.

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From Wikipedia:

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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.

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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

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Usage

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Example:

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

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Note

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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.

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++ellps=<value>
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The name of a built-in ellipsoid definition.

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

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Defaults to “GRS80”.

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++lon_0=<value>
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Longitude of projection center.

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Defaults to 0.0.

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++x_0=<value>
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False easting.

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Defaults to 0.0.

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++y_0=<value>
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False northing.

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Defaults to 0.0.

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Mathematical definition

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

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Forward projection

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

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

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  1. Wikipedia

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