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| author | Kristian Evers <kristianevers@gmail.com> | 2018-02-28 22:37:13 +0100 |
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| committer | Kristian Evers <kristianevers@gmail.com> | 2018-02-28 22:37:13 +0100 |
| commit | be3791ffd5e802d5a3d38fa08f5ed24715b73c7c (patch) | |
| tree | 25eff60d91fed8ffce3302e37818f5bc452b16ca /docs/source/operations/projections/merc.rst | |
| parent | db2bfdce6df26801584be562f0ac74648ef8cefd (diff) | |
| download | PROJ-be3791ffd5e802d5a3d38fa08f5ed24715b73c7c.tar.gz PROJ-be3791ffd5e802d5a3d38fa08f5ed24715b73c7c.zip | |
Move 'Coordinate operations' to top level of docs [skip ci]
Diffstat (limited to 'docs/source/operations/projections/merc.rst')
| -rw-r--r-- | docs/source/operations/projections/merc.rst | 140 |
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diff --git a/docs/source/operations/projections/merc.rst b/docs/source/operations/projections/merc.rst new file mode 100644 index 00000000..8895abc8 --- /dev/null +++ b/docs/source/operations/projections/merc.rst @@ -0,0 +1,140 @@ +.. _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>`_ + + |
