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.. _webmerc:
********************************************************************************
Web Mercator / Pseudo Mercator
********************************************************************************
.. versionadded:: 5.1.0
The Web Mercator / Pseudo Mercator projection is a cylindrical map projection.
This is a variant of the regular :ref:`merc` projection, except that the computation
is done on a sphere, using the semi-major axis of the ellipsoid.
From `Wikipedia <https://en.wikipedia.org/wiki/Web_Mercator>`_:
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.
.. include:: ../options/R.rst
.. include:: ../options/x_0.rst
.. include:: ../options/y_0.rst
Mathematical definition
#######################
The formulas describing the Mercator projection are all taken from G. Evenden's libproj manuals [Evenden2005]_.
Forward projection
==================
.. math::
x = \lambda
.. math::
y = \ln \left[ \tan \left(\frac{\pi}{4} + \frac{\phi}{2} \right) \right]
Inverse projection
==================
.. math::
\lambda = {x}
.. math::
\phi = \frac{\pi}{2} - 2 \arctan \left[ e^{-y} \right]
Further reading
###############
#. `Wikipedia <https://en.wikipedia.org/wiki/Web_Mercator>`_
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