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.. _mill:
********************************************************************************
Miller Cylindrical
********************************************************************************
The Miller cylindrical projection is a modified Mercator projection, proposed by Osborn Maitland Miller in 1942.
The latitude is scaled by a factor of :math:`\frac{4}{5}`, projected according to Mercator, and then the result is multiplied by :math:`\frac{5}{4}` to retain scale along the equator.
+---------------------+----------------------------------------------------------+
| **Classification** | Neither conformal nor equal area cylindrical |
+---------------------+----------------------------------------------------------+
| **Available forms** | Forward and inverse spherical |
+---------------------+----------------------------------------------------------+
| **Defined area** | Global, but best used near the equator |
+---------------------+----------------------------------------------------------+
| **Alias** | mill |
+---------------------+----------------------------------------------------------+
| **Domain** | 2D |
+---------------------+----------------------------------------------------------+
| **Input type** | Geodetic coordinates |
+---------------------+----------------------------------------------------------+
| **Output type** | Projected coordinates |
+---------------------+----------------------------------------------------------+
.. image:: ./images/mill.png
:scale: 50%
:alt: Miller Cylindrical
Usage
########
The Miller Cylindrical projection is used for world maps and in several atlases,
including the National Atlas of the United States (USGS, 1970, p. 330-331) :cite:`Snyder1987`.
Example using Central meridian 90°W::
$ echo -100 35 | proj +proj=mill +lon_0=90w
-1113194.91 4061217.24
Parameters
################################################################################
.. note:: All parameters for the projection are optional.
.. include:: ../options/lon_0.rst
.. include:: ../options/R.rst
.. include:: ../options/x_0.rst
.. include:: ../options/y_0.rst
Mathematical definition
#######################
The formulas describing the Miller projection are all taken from :cite:`Snyder1987`.
Forward projection
==================
.. math::
x = \lambda
.. math::
y = 1.25 * \ln \left[ \tan \left(\frac{\pi}{4} + 0.4 * \phi \right) \right]
Inverse projection
==================
.. math::
\lambda = x
.. math::
\phi = 2.5 * ( \arctan \left[ e^{0.8 * y} \right] - \frac{\pi}{4} )
Further reading
###############
#. `Wikipedia <https://en.wikipedia.org/wiki/Miller_cylindrical_projection>`_
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