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.. _merc:
********************************************************************************
Mercator
********************************************************************************
The Mercator projection is a cylindrical map projection that origins
from the 16th century. It is widely recognized as the first regularly
used map projection. It is a conformal projection in which the equator
projects to a straight line at constant scale. The projection has the
property that a rhumb line, a course of constant heading, projects to a
straight line. This makes it suitable for navigational purposes.
+---------------------+----------------------------------------------------------+
| **Classification** | Conformal cylindrical |
+---------------------+----------------------------------------------------------+
| **Available forms** | Forward and inverse, spherical and ellipsoidal |
+---------------------+----------------------------------------------------------+
| **Defined area** | Global, but best used near the equator |
+---------------------+----------------------------------------------------------+
| **Alias** | merc |
+---------------------+----------------------------------------------------------+
| **Domain** | 2D |
+---------------------+----------------------------------------------------------+
| **Input type** | Geodetic coordinates |
+---------------------+----------------------------------------------------------+
| **Output type** | Projected coordinates |
+---------------------+----------------------------------------------------------+
.. figure:: ./images/merc.png
:width: 500 px
:align: center
:alt: Mercator
proj-string: ``+proj=merc``
Usage
########
Applications should be limited to equatorial regions, but is frequently
used for navigational charts with latitude of true scale (:option:`+lat_ts`) specified within
or near chart's boundaries.
It is considered to be inappropriate for world maps because of the gross
distortions in area; for example the projected area of Greenland is
larger than that of South America, despite the fact that Greenland's
area is :math:`\frac18` that of South America :cite:`Snyder1987`.
Example using latitude of true scale::
$ echo 56.35 12.32 | proj +proj=merc +lat_ts=56.5
3470306.37 759599.90
Example using scaling factor::
echo 56.35 12.32 | proj +proj=merc +k_0=2
12545706.61 2746073.80
Note that :option:`+lat_ts` and :option:`+k_0` are mutually exclusive.
If used together, :option:`+lat_ts` takes precedence over :option:`+k_0`.
Parameters
################################################################################
.. note:: All parameters for the projection are optional.
.. include:: ../options/lat_ts.rst
.. include:: ../options/k_0.rst
.. include:: ../options/lon_0.rst
.. include:: ../options/x_0.rst
.. include:: ../options/y_0.rst
.. include:: ../options/ellps.rst
.. include:: ../options/R.rst
Mathematical definition
#######################
Spherical form
**************
For the spherical form of the projection we introduce the scaling factor:
.. math::
k_0 = \cos \phi_{ts}
Forward projection
==================
.. math::
x = k_0R \lambda; \qquad y = k_0R \psi
.. math::
\psi &= \ln \tan \biggl(\frac{\pi}{4} + \frac{\phi}{2} \biggr)\\
&= \sinh^{-1}\tan\phi
The quantity :math:`\psi` is the isometric latitude.
Inverse projection
==================
.. math::
\lambda = \frac{x}{k_0R}; \qquad \psi = \frac{y}{k_0R}
.. math::
\phi &= \frac{\pi}{2} - 2 \tan^{-1} \exp(-\psi)\\
&= \tan^{-1}\sinh\psi
Ellipsoidal form
****************
For the ellipsoidal form of the projection we introduce the scaling factor:
.. math::
k_0 = m( \phi_{ts} )
where
.. math::
m(\phi) = \frac{\cos\phi}{\sqrt{1 - e^2\sin^2\phi}}
:math:`a\,m(\phi)` is the radius of the circle of latitude :math:`\phi`.
Forward projection
==================
.. math::
x = k_0 a \lambda; \qquad y = k_0 a \psi
.. math::
\psi &= \ln\tan\biggl(\frac\pi4 + \frac{\phi}2\biggr)
-\frac12 e
\ln \biggl(\frac{1 + e \sin\phi}{1 - e \sin\phi}\biggr)\\
&= \sinh^{-1}\tan\phi - e \tanh^{-1}(e \sin\phi)
Inverse projection
==================
.. math::
\lambda = \frac{x}{k_0 a}; \quad \psi = \frac{y}{k_0 a}
The latitude :math:`\phi` is found by inverting the equation for
:math:`\psi` iteratively.
Further reading
###############
#. `Wikipedia <https://en.wikipedia.org/wiki/Mercator_projection>`_
#. `Wolfram Mathworld <http://mathworld.wolfram.com/MercatorProjection.html>`_
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