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authorKristian Evers <kristianevers@gmail.com>2018-02-28 22:37:13 +0100
committerKristian Evers <kristianevers@gmail.com>2018-02-28 22:37:13 +0100
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+.. _merc:
+
+********************************************************************************
+Mercator
+********************************************************************************
+
+The Mercator projection is a cylindrical map projection that origins from the 15th
+century. It is widely recognized as the first regularly used map projection.
+The projection is conformal which makes it suitable for navigational purposes.
+
+
++---------------------+----------------------------------------------------------+
+| **Classification** | Conformal cylindrical |
++---------------------+----------------------------------------------------------+
+| **Available forms** | Forward and inverse, spherical and elliptical projection |
++---------------------+----------------------------------------------------------+
+| **Defined area** | Global, but best used near the equator |
++---------------------+----------------------------------------------------------+
+| **Implemented by** | Gerald I. Evenden |
++---------------------+----------------------------------------------------------+
+| **Options** |
++---------------------+----------------------------------------------------------+
+| `+lat_ts` | Latitude of true scale. Defaults to 0.0 |
++---------------------+----------------------------------------------------------+
+| `+k_0` | Scaling factor. Defaults to 1.0 |
++---------------------+----------------------------------------------------------+
+
+
+
+.. image:: ./images/merc.png
+ :scale: 50%
+ :alt: Mercator
+
+
+Usage
+########
+
+Applications should be limited to equatorial regions, but is frequently
+used for navigational charts with latitude of true scale (``+lat_ts``) specified within
+or near chart's boundaries.
+Often inappropriately used for world maps since the regions near the poles
+cannot be shown [Evenden1995]_.
+
+
+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 ``+lat_ts`` and ``+k_0`` are mutually exclusive.
+If used together, ``+lat_ts`` takes precedence over ``+k_0``.
+
+Mathematical definition
+#######################
+
+The formulas describing the Mercator projection are all taken from G. Evenden's libproj manuals [Evenden2005]_.
+
+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_0 \lambda
+
+.. math::
+
+ y = k_0 \ln \left[ \tan \left(\frac{\pi}{4} + \frac{\phi}{2} \right) \right]
+
+
+Inverse projection
+==================
+
+.. math::
+
+ \lambda = \frac{x}{k_0}
+
+.. math::
+
+ \phi = \frac{\pi}{2} - 2 \arctan \left[ e^{-y/k_0} \right]
+
+
+Elliptical form
+***************
+
+For the elliptical form of the projection we introduce the scaling factor:
+
+.. math::
+
+ k_0 = m\left( \phi_ts \right)
+
+where :math:`m\left(\phi\right)` is the parallel radius at latitude :math:`\phi`.
+
+We also use the Isometric Latitude kernel function :math:`t()`.
+
+.. note::
+ m() and t() should be described properly on a separate page about the theory of projections on the ellipsoid.
+
+Forward projection
+==================
+.. math::
+
+ x = k_0 \lambda
+
+.. math::
+
+ y = k_0 \ln t \left( \phi \right)
+
+
+Inverse projection
+==================
+
+.. math::
+
+ \lambda = \frac{x}{k_0}
+
+.. math::
+
+ \phi = t^{-1}\left[ e^{ -y/k_0 } \right]
+
+Further reading
+###############
+
+#. `Wikipedia <https://en.wikipedia.org/wiki/Mercator_projection>`_
+#. `Wolfram Mathworld <http://mathworld.wolfram.com/MercatorProjection.html>`_
+
+