.. _mill:

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Miller Cylindrical
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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                                         |
+---------------------+--------------------------------------------------------------------------------+
| **Implemented by**  | Gerald I. Evenden                                                              |
+---------------------+--------------------------------------------------------------------------------+
| **Options**                                                                                          |
+---------------------+--------------------------------------------------------------------------------+
| `+lat_0`            | Latitude of origin (Default to 0)                                              |
+---------------------+--------------------------------------------------------------------------------+

.. 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) [Snyder1987]_.

Example using Central meridian 90°W::

    $ echo -100 35 | proj +proj=mill +lon_0=90w
    -1113194.91      4061217.24

Mathematical definition
#######################

The formulas describing the Miller projection are all taken from Snyder's manuals [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>`_