GMT Map Projections
===================

GMT implements more than 30 different projections. They all project the input coordinates longitude and latitude to
positions on a map. In general, :math:`x' = f(x,y,z)` and :math:`y' = g(x,y,z)`, where :math:`z` is implicitly given as
the radial vector length to the :math:`(x,y)` point on the chosen ellipsoid. The functions :math:`f` and :math:`g` can be
quite nasty and we will refrain from presenting details in this document. The interested reader is referred to *Snyder*
[1987]\ [20]_. We will mostly be using the :doc:`/coast` command to demonstrate each of the projections. GMT map
projections are grouped into four categories depending on the nature of the projection. The groups are

#. :ref:`cookbook/map-projections:Conic projections`

#. :ref:`cookbook/map-projections:Azimuthal projections`

#. :ref:`cookbook/map-projections:Cylindrical projections`

#. :ref:`cookbook/map-projections:Miscellaneous projections`

Because :math:`x` and :math:`y` are coupled we can only specify one plot-dimensional scale, typically a map *scale*
(for lower-case map projection code) or a map *width* (for upper-case map projection code). The *measurement unit*
is cm, inch, or point, depending on the :term:`PROJ_LENGTH_UNIT` setting in **gmt.conf**, but this can be overridden on
the command line by appending **c**, **i**, or **p** to the *scale* or *width* values. In some cases it would be more
practical to specify map *height* instead of *width*, while in other situations it would be nice to set either the
*shortest* or *longest* map dimension. Users may select these alternatives by appending a character code to their map
dimension [detault is **+dw**]:

   - Append **+dh** to the given :ref:`dimension <cookbook/features:Dimension units>` to specify map *height*.
   - Append **+du** to the given :ref:`dimension <cookbook/features:Dimension units>` to select the minimum map
     dimension.
   - Append **+dl** to the given :ref:`dimension <cookbook/features:Dimension units>` to select the maximum map
     dimension.

The ellipsoid used in map projections is user-definable. 73 commonly used ellipsoids and spheroids are currently
supported, and users may also specify their own custom ellipsoid parameters [default is WGS-84]. Several GMT parameters
can affect the projection: :term:`PROJ_ELLIPSOID`, :term:`GMT_INTERPOLANT`, :term:`PROJ_SCALE_FACTOR`, and
:term:`PROJ_LENGTH_UNIT`; see the :doc:`../gmt.conf` man page for details.

In GMT version 4.3.0 we noticed we ran out of the alphabet for 1-letter (and sometimes 2-letter) projection codes. To
allow more flexibility, and to make it easier to remember the codes, we implemented the option to use the abbreviations
used by the `PROJ <https://proj.org/>`_ mapping package. Since some of the GMT projections are not in **PROJ**, we
invented some of our own as well. For a full list of both the old 1- and 2-letter codes, as well as the
**PROJ**-equivalents see the quick reference table below. For example, **-JM**\ 15c and **-JMerc**\ /15c have the same
meaning.

.. include:: ../proj-codes.rst_

Conic projections
-----------------

.. _-Jb:

Albers conic equal-area projection (**-Jb** **-JB**)
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

**Syntax**

    **-Jb**\|\ **B**\ *lon0/lat0/lat1/lat2/*\ *scale*\|\ *width*

**Parameters**

- The longitude (*lon0*) and latitude (*lat0*) of the projection center.
- The two standard parallels (*lat1* and *lat2*).
- The *scale* in :ref:`plot-units <plt-units>`/degree or as 1:xxxxx (with **-Jb**) or map *width* in
  :ref:`plot-units <plt-units>` (with **-JB**).

Note that you must include the "1:" if you choose to specify the *scale* that way. For example, you can say 0.5c which
means 0.5 cm/degree or 1:200000 which means 1 unit on the map equals 200,000 units along the standard parallels. The
projection center defines the origin of the rectangular map coordinates.

**Description**

This projection, developed by Heinrich C. Albers in 1805, is predominantly used to map regions of large east-west
extent, in particular the United States. It is a conic, equal-area projection, in which parallels are unequally
spaced arcs of concentric circles, more closely spaced at the north and south edges of the map. Meridians, on the other
hand, are equally spaced radii about a common center, and cut the parallels at right angles. Distortion in scale and
shape vanishes along the two standard parallels. Between them, the scale along parallels is too small; beyond them it is
too large. The opposite is true for the scale along meridians.

**Example**

As an example we will make a map of the region near Taiwan. We choose the center of the projection to be at 125°E/20°N
and 25°N and 45°N as our two standard parallels. We desire a map that is 12 cm wide. The complete command needed to
generate the map below is therefore given by:

.. literalinclude:: /_verbatim/GMT_albers.txt

.. figure:: /_images/GMT_albers.*
   :width: 500 px
   :align: center

   Albers equal-area conic map projection.

.. _-Jd:

Equidistant conic projection (**-Jd** **-JD**)
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

**Syntax**

    **-Jd**\|\ **D**\ *lon0/lat0/lat1/lat2/*\ *scale*\|\ *width*

**Parameters**

- The longitude (*lon0*) and latitude (*lat0*) of the projection center.
- The two standard parallels (*lat1* and *lat2*).
- The *scale* in :ref:`plot-units <plt-units>`/degree or as 1:xxxxx (with **-Jd**) or map *width* in
  :ref:`plot-units <plt-units>` (with **-JD**).

**Description**

The equidistant conic projection was described by the Greek philosopher Claudius Ptolemy about A.D. 150. It is neither
conformal or equal-area, but serves as a compromise between them. The scale is true along all meridians and the standard
parallels.

**Example**

The equidistant conic projection is often used for atlases with maps of small countries. As an example, we generate a
map of Cuba:

.. literalinclude:: /_verbatim/GMT_equidistant_conic.txt

.. figure:: /_images/GMT_equidistant_conic.*
   :width: 500 px
   :align: center

   Equidistant conic map projection.

.. _-Jl:

Lambert conic conformal projection (**-Jl** **-JL**)
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

**Syntax**

    **-Jl**\|\ **L**\ *lon0/lat0/lat1/lat2/*\ *scale*\|\ *width*

**Parameters**

- The longitude (*lon0*) and latitude (*lat0*) of the projection center.
- The two standard parallels (*lat1* and *lat2*).
- The *scale* in :ref:`plot-units <plt-units>`/degree or as 1:xxxxx (with **-Jl**) or map *width* in
  :ref:`plot-units <plt-units>` (with **-JL**).

**Description**

This conic projection was designed by the Alsatian mathematician Johann Heinrich Lambert (1772) and has been used
extensively for mapping of regions with predominantly east-west orientation, just like the Albers projection. Unlike the
Albers projection, Lambert's conformal projection is not equal-area. The parallels are arcs of circles with a common
origin, and meridians are the equally spaced radii of these circles. As with Albers projection, it is only the two
standard parallels that are distortion-free.

**Example**

The Lambert conformal projection has been used for basemaps for all the 48 contiguous States with the two fixed standard
parallels 33°N and 45°N. We will generate a map of the continental USA using these parameters. Note that with all the
projections you have the option of selecting a rectangular border rather than one defined by meridians and parallels.
Here, we choose the regular WESN region, a "fancy" basemap frame, and use degrees west for longitudes. The generating
commands used were:

.. literalinclude:: /_verbatim/GMT_lambert_conic.txt

.. figure:: /_images/GMT_lambert_conic.*
   :width: 500 px
   :align: center

   Lambert conformal conic map projection.


The choice for projection center does not affect the projection but it indicates which meridian (here 100°W) will be
vertical on the map. The standard parallels were originally selected by Adams to provide a maximum scale error between
latitudes 30.5°N and 47.5°N of 0.5–1%. Some areas, like Florida, experience scale errors of up to 2.5%.

.. _-Jpoly:

(American) polyconic projection (**-Jpoly** **-JPoly**)
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

**Syntax**

    **-Jpoly**\|\ **Poly**\ /[*lon0/*\ [*lat0/*]]\ *scale*\|\ *width*

**Parameters**

- The longitude (*lon0*) and latitude (*lat0*) of the projection center.
- The two standard parallels (*lat1* and *lat2*).
- The *scale* in :ref:`plot-units <plt-units>`/degree or as 1:xxxxx (with **-Jl**) or map *width* in
  :ref:`plot-units <plt-units>` (with **-JL**).

**Description**

The polyconic projection, in Europe usually referred to as the American polyconic projection, was introduced shortly
before 1820 by the Swiss-American cartographer Ferdinand Rodulph Hassler (1770–1843). As head of the Survey of the
Coast, he was looking for a projection that would give the least distortion for mapping the coast of the United
States. The projection acquired its name from the construction of each parallel, which is achieved by projecting the
parallel onto the cone while it is rolled around the globe, along the central meridian, tangent to that parallel. As a
consequence, the projection involves many cones rather than a single one used in regular conic projections.

The polyconic projection is neither equal-area, nor conformal. It is true to scale without distortion along the central
meridian. Each parallel is true to scale as well, but the meridians are not as they get further away from the central
meridian. As a consequence, no parallel is standard because conformity is lost with the lengthening of the meridians.

**Example**

Below we reproduce the illustration by *Snyder* [1987], with a gridline every 10 and annotations only every 30° in
longitude:

.. literalinclude:: /_verbatim/GMT_polyconic.txt

.. figure:: /_images/GMT_polyconic.*
   :width: 500 px
   :align: center

   (American) polyconic projection.


Azimuthal projections
---------------------

.. _-Ja:

Lambert Azimuthal Equal-Area (**-Ja** **-JA**)
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

**Syntax**

    **-Ja**\|\ **A**\ *lon0/lat0*\ [*/horizon*]\ *scale*\|\ *width*

**Parameters**

- The longitude (*lon0*) and latitude (*lat0*) of the projection center.
- Optionally, the *horizon*, i.e., the number of degrees from the center to the edge (<=180) [default is 90].
- The *scale* as 1:xxxxx or as *radius/latitude* where *radius* is the projected distance on the map from projection center
  to an oblique latitude where 0 would be the oblique Equator (with **-Ja**) or map *width*
  :ref:`plot-units <plt-units>` (with **-JA**).

**Description**

This projection was developed by Johann Heinrich Lambert in 1772 and is typically used for mapping large regions like
continents and hemispheres. It is an azimuthal, equal-area projection, but is not perspective. Distortion is zero at the
center of the projection, and increases radially away from this point.

**Examples**

Two different types of maps can be made with this projection depending on how the region is specified. We will give
examples of both types in the next two subsections.

Rectangular map
^^^^^^^^^^^^^^^

In this mode we define our region by specifying the longitude/latitude
of the lower left and upper right corners instead of the usual *west,
east, south, north* boundaries. The reason for specifying our area this
way is that for this and many other projections, lines of equal
longitude and latitude are not straight lines and are thus poor choices
for map boundaries. Instead we require that the map boundaries be
rectangular by defining the corners of a rectangular map boundary. Using
0°E/40°S (lower left) and 60°E/10°S (upper right) as our corners we try

.. literalinclude:: /_verbatim/GMT_lambert_az_rect.txt

.. figure:: /_images/GMT_lambert_az_rect.*
   :width: 500 px
   :align: center

   Rectangular map using the Lambert azimuthal equal-area projection.


Note that an **+r** is appended to the **-R** option to inform GMT that
the region has been selected using the rectangle technique, otherwise it
would try to decode the values as *west, east, south, north* and report
an error since *'east'* < *'west'*.

Hemisphere map
^^^^^^^^^^^^^^

Here, you must specify the world as your region (**-Rg** or **-Rd**). E.g., to obtain a hemisphere view that shows the
Americas, try

.. literalinclude:: /_verbatim/GMT_lambert_az_hemi.txt

.. figure:: /_images/GMT_lambert_az_hemi.*
   :width: 400 px
   :align: center

   Hemisphere map using the Lambert azimuthal equal-area projection.


To geologists, the Lambert azimuthal equal-area projection (with origin
at 0/0) is known as the *equal-area* (Schmidt) stereonet and used for
plotting fold axes, fault planes, and the like. An *equal-angle* (Wulff)
stereonet can be obtained by using the stereographic projection
(discussed later). The stereonets produced by these two projections appear below.

.. _GMT_stereonets:

.. figure:: /_images/GMT_stereonets.*
   :width: 500 px
   :align: center

   Equal-Area (Schmidt) and Equal-Angle (Wulff) stereo nets.

.. toggle::

   Here is the source script for the figure above:

   .. literalinclude:: /_verbatim/GMT_stereonets.txt


.. _-Js:

Stereographic Equal-Angle (**-Js** **-JS**)
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

**Syntax**

    **-Js**\|\ **S**\ *lon0/lat0*\ [*/horizon*]\ */*\ *scale*\|\ *width*

**Parameters**

- The longitude (*lon0*) and latitude (*lat0*) of the projection center.
- Optionally, the *horizon*, i.e., the number of degrees from the center to the edge (< 180) [default is 90].
- Scale (via **-Js**) may be provided in one of three flavors: Append 1:xxxxx (true scale at pole), *slat*\ /1:xxxxx
  (true scale at standard parallel *slat* for the polar aspect only), or *radius/latitude* (where *radius* is
  distance on map in :ref:`plot-units <plt-units>` from projection center to a particular oblique latitude).
- Alternatively, simply append map *width* in :ref:`plot-units <plt-units>` (with **-JS**).

**Description**


This is a conformal, azimuthal projection that dates back to the Greeks. Its main use is for mapping the polar regions.
In the polar aspect all meridians are straight lines and parallels are arcs of circles. While this is the most common
use it is possible to select any point as the center of projection.

A map scale factor of 0.9996 will be applied by default (although you may change this with :term:`PROJ_SCALE_FACTOR`).
However, the setting is ignored when a standard parallel has been specified since the scale is then implicitly given.
We will look at two different types of maps.

**Examples**

Multiple types of maps can be made with this projection depending on how the region is specified. We will give
examples in the next three subsections.

Polar Stereographic Map
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

In our first example we will let the projection center be at the north
pole. This means we have a polar stereographic projection and the map
boundaries will coincide with lines of constant longitude and latitude.
An example is given by:

.. literalinclude:: /_verbatim/GMT_stereographic_polar.txt

.. figure:: /_images/GMT_stereographic_polar.*
   :width: 500 px
   :align: center

   Polar stereographic conformal projection.


Rectangular stereographic map
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

As with Lambert's azimuthal equal-area projection we have the option to
use rectangular boundaries rather than the wedge-shape typically
associated with polar projections. This choice is defined by selecting
two points as corners in the rectangle and appending **+r** to the
**-R** option. This command produces a map as presented in
Figure :ref:`Polar stereographic <GMT_stereographic_rect>`:

.. literalinclude:: /_verbatim/GMT_stereographic_rect.txt

.. _GMT_stereographic_rect:

.. figure:: /_images/GMT_stereographic_rect.*
   :width: 500 px
   :align: center

   Polar stereographic conformal projection with rectangular borders.


General stereographic map
^^^^^^^^^^^^^^^^^^^^^^^^^

In terms of usage this projection is identical to the Lambert azimuthal
equal-area projection. Thus, one can make both rectangular and
hemispheric maps. Our example shows Australia using a projection pole at
130°E/30°S. The command used was

.. literalinclude:: /_verbatim/GMT_stereographic_general.txt

.. figure:: /_images/GMT_stereographic_general.*
   :width: 500 px
   :align: center

   General stereographic conformal projection with rectangular borders.


By choosing 0/0 as the pole, we obtain the conformal stereonet presented
next to its equal-area cousin in the Section `Lambert Azimuthal Equal-Area (-Ja -JA)`_ on the Lambert azimuthal equal-area projection (Figure :ref:`Stereonets <GMT_stereonets>`).

.. _-Jg_pers:

Perspective projection (**-Jg** **-JG**)
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

**Syntax**

    **-Jg**\|\ **G**\ *lon0/lat0*\ */*\ *scale*\|\ *width*\ [**+a**\ *azimuth*][**+t**\ *tilt*][**+v**\ *vwidth/vheight*][**+w**\ *twist*][**+z**\ *altitude*\ [**r**\|\ **R**]\|\ **g**]

**Required Parameters**

- The longitude (*lon0*) and latitude (*lat0*) of the projection center.
- The *scale* as 1:xxxxx or as *radius/latitude* where *radius* is distance on map in :ref:`plot-units <plt-units>` from
  projection center to a particular oblique latitude (with **-Jg**), or map width in :ref:`plot-units <plt-units>`
  (with **-JG**).

**Optional Parameters**

- The *azimuth* in degrees. This is the direction in which you are looking, measured clockwise from north [0].
- The *tilt* in degrees. This is the viewing angle relative to zenith. For example, a tilt of 0° is looking straight
  down, and 60° is looking from 30° above the horizon [0].
- The *vwidth* and *vheight* of the viewpoint in degrees. This number depends on whether you are looking with the naked
  eye (in which case the view is about 60° wide), or with binoculars, for example [unrestricted].
- The *twist* in degrees. This is the boresight rotation (clockwise) of the image [0].
- The *altitude* of the viewer above sea level in kilometers [infinity]. Alternatively, append **R** if giving
  the distance of the viewer from the center of the Earth in Earth radii, or **r** if giving the distance from the
  center of the Earth in kilometers.  Finally, give *altitude* as **g** to compute and use the altitude for a
  geosynchronous orbit.

**Description**

The perspective projection imitates in 2 dimensions the 3-dimensional view of the earth from space. The implementation
in GMT is very flexible, and thus requires many input variables.

**Example**

The imagined view of northwest Europe from a Space Shuttle at 230 km looking due east is thus accomplished by the
following :doc:`/coast` command (*lon0*\ =4; *lat0*\ =52; *altitude*\ =230 km; *azimuth*\ = 90°; *tilt*\ = 60°;
*twist*\ = 180°; *vwidth*\ = *vheight*\ = 60°; and *width* = 12 cm):

.. literalinclude:: /_verbatim/GMT_perspective.txt

.. _GMT_perspective:

.. figure:: /_images/GMT_perspective.*
   :width: 500 px
   :align: center

   View from the Space Shuttle in Perspective projection.

.. _-Jg:

Orthographic projection (**-Jg** **-JG**)
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

**Syntax**

    **-Jg**\|\ **G**\ *lon0/lat0*\ [*/horizon*]\ */*\ *scale*\|\ *width*

**Parameters**

- The longitude (*lon0*) and latitude (*lat0*) of the projection center.
- Optionally, the *horizon*, i.e., the number of degrees from the center to the edge (<=90) [default is 90].
- The *scale* as 1:xxxxx or as *radius/latitude* where *radius* is distance on map in :ref:`plot-units <plt-units>` from
  projection center to a particular oblique latitude (with **-Jg**), or map width in :ref:`plot-units <plt-units>`
  (with **-JG**).

**Description**

The orthographic azimuthal projection is a perspective projection from infinite distance. It is therefore often used to
give the appearance of a globe viewed from outer space. As with Lambert's equal-area and the stereographic projection,
only one hemisphere can be viewed at any time. The projection is neither equal-area nor conformal, and much distortion
is introduced near the edge of the hemisphere. The directions from the center of projection are true. The projection was
known to the Egyptians and Greeks more than 2,000 years ago. Because it is mainly used for pictorial views at a small
scale, only the spherical form is necessary.

**Example**

Our example of a perspective view centered on 75°W/40°N can therefore be generated by the following :doc:`/coast`
command:

.. literalinclude:: /_verbatim/GMT_orthographic.txt

.. figure:: /_images/GMT_orthographic.*
   :width: 400 px
   :align: center

   Hemisphere map using the Orthographic projection.

.. _-Je:

Azimuthal Equidistant projection (**-Je** **-JE**)
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

**Syntax**

    **-Je**\|\ **E**\ *lon0/lat0*\ [*/horizon*]\ *scale*\|\ *width*

**Parameters**

- The longitude (*lon0*) and latitude (*lat0*) of the projection center.
- Optionally, the *horizon*, i.e., the number of degrees from the center to the edge (<=180) [default is 180].
- The *scale* as 1:xxxxx or as *radius/latitude* where *radius* is distance on map in :ref:`plot-units <plt-units>` from
  projection center to a particular oblique latitude (with **-Je**), or map width in :ref:`plot-units <plt-units>`
  (with **-JE**).

**Description**

The most noticeable feature of this azimuthal projection is the fact that distances measured from the center are true.
Therefore, a circle about the projection center defines the locus of points that are equally far away from the plot
origin. Furthermore, directions from the center are also true. The projection, in the polar aspect, is at least several
centuries old. It is a useful projection for a global view of locations at various or identical distance from a given
point (the map center).

**Example**

Our example of a global view centered on 100°W/40°N can therefore be generated by the following :doc:`/coast` command.
Note that the antipodal point is 180° away from the center, but in this projection this point plots as the entire map
perimeter:

.. literalinclude:: /_verbatim/GMT_az_equidistant.txt

.. figure:: /_images/GMT_az_equidistant.*
   :width: 400 px
   :align: center

   World map using the equidistant azimuthal projection.

.. _-Jf:

Gnomonic projection (**-Jf** **-JF**)
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

**Syntax**

    **-Jf**\|\ **F**\ *lon0/lat0*\ [*/horizon*]\ *scale*\|\ *width*

**Parameters**

- The longitude (*lon0*) and latitude (*lat0*) of the projection center.
- Optionally, the *horizon*, i.e., the number of degrees from the center to the edge (<90) [default is 60].
- The *scale* as 1:xxxxx or as *radius/latitude* where radius is distance on map in :ref:`plot-units <plt-units>` from
  projection center to a particular oblique latitude (with **-Jf**), or map width in :ref:`plot-units <plt-units>`
  (with **-JF**).

**Description**

The Gnomonic azimuthal projection is a perspective projection from the center onto a plane tangent to the surface. Its
origin goes back to the old Greeks who used it for star maps almost 2500 years ago. The projection is neither equal-area
nor conformal, and much distortion is introduced near the edge of the hemisphere; in fact, less than a hemisphere may be
shown around a given center. The directions from the center of projection are true. Great circles project onto straight
lines. Because it is mainly used for pictorial views at a small scale, only the spherical form is necessary.

**Example**

Using a *horizon* of 60, our example of this projection centered on 120°W/35°N can therefore be generated by the
following :doc:`/coast` command:

.. literalinclude:: /_verbatim/GMT_gnomonic.txt

.. figure:: /_images/GMT_gnomonic.*
   :width: 500 px
   :align: center

   Gnomonic azimuthal projection.


Cylindrical projections
-----------------------

Cylindrical projections are easily recognized for their shape: maps are rectangular and meridians and parallels are
straight lines crossing at right angles. But that is where similarities between the cylindrical projections supported
by GMT (:ref:`Mercator <-Jm>`, :ref:`transverse Mercator <-Jt>`, :ref:`universal transverse Mercator <-Ju>`,
:ref:`oblique Mercator <-Jo>`, :ref:`Cassini <-Jc>`, :ref:`cylindrical equidistant <-Jq>`,
:ref:`cylindrical equal-area <-Jy>`, :ref:`Miller <-Jj>`, and :ref:`cylindrical stereographic <-Jcyl_stere>`) stops.
Each have a different way of spacing the meridians and parallels to obtain certain desirable cartographic properties.

.. _-Jm:

Mercator projection (**-Jm** **-JM**)
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

**Syntax**

   **-Jm**\|\ **M**\ [*lon0/*\ [*lat0/*]]\ *scale*\|\ *width*

**Parameters**

-  Optionally, the central meridian (*lon0*) [default is the middle of the map].
-  Optionally, the standard parallel for true scale (*lat0*) [default is the equator]. When supplied, the central
   meridian (*lon0*) must be supplied as well.
-  The *scale* along the equator in :ref:`plot-units <plt-units>`/degree or as 1:xxxxx (with **-Jm**) or map *width* in
   :ref:`plot-units <plt-units>` (with **-JM**).

**Description**

Probably the most famous of the various map projections, the Mercator projection takes its name from the Flemish
cartographer Gheert Cremer, better known as Gerardus Mercator, who presented it in 1569. The projection is a cylindrical
and conformal, with no distortion along the equator. A major navigational feature of the projection is that a line of
constant azimuth is straight. Such a line is called a rhumb line or *loxodrome*. Thus, to sail from one point to another
one only had to connect the points with a straight line, determine the azimuth of the line, and keep this constant
course for the entire voyage\ [21]_. The Mercator projection has been used extensively for world maps in which the
distortion towards the polar regions grows rather large, thus incorrectly giving the impression that, for example,
Greenland is larger than South America. In reality, the latter is about eight times the size of Greenland. Also, the
Former Soviet Union looks much bigger than Africa or South America. One may wonder whether this illusion has had any
influence on U.S. foreign policy.

In the regular Mercator projection, the cylinder touches the globe along the equator. Other orientations like vertical
and oblique give rise to the :ref:`transverse Mercator <-Jt>` and :ref:`oblique Mercator <-Jo>` projections,
respectively. We will discuss these generalizations following the regular Mercator projection.

**Example**

A world map at a scale of 0.03 cm per degree, which will give a map 10.8-cm wide, can be obtained as follows:

.. literalinclude:: /_verbatim/GMT_mercator.txt

.. figure:: /_images/GMT_mercator.*
   :width: 500 px
   :align: center

   Simple Mercator map.


While this example is centered on the Dateline, one can easily choose another configuration with the **-R** option. For
example, specify the region with **-R**-180/180/-70/70 to obtain a map centered on Greenwich.

.. _-Jt:

Transverse Mercator projection (**-Jt** **-JT**)
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

**Syntax**

   **-Jt**\|\ **T**\ *lon0/*\ [*lat0/*]\ *scale*\|\ *width*

**Parameters**

-  The central meridian (*lon0*).
-  Optionally, the latitude of origin (*lat0*) [default is the equator].
-  The *scale* along the equator in :ref:`plot-units <plt-units>`/degree or 1:xxxxx (with **-Jt**) or map
   *width* in :ref:`plot-units <plt-units>` (with **-JT**).

You can change the map scale factor via the :term:`PROJ_SCALE_FACTOR` parameter [default is **1**].

**Description**

The transverse Mercator was invented by Johann Heinrich Lambert in 1772. In this projection the cylinder touches a
meridian along which there is no distortion. The distortion increases away from the central meridian and goes to
infinity at 90° from center. The central meridian, each meridian 90° away from the center, and equator are straight
lines; other parallels and meridians are complex curves.

**Example**

A transverse Mercator map of south-east Europe and the Middle East with 35°E as the central meridian can be obtained as
follows:

.. literalinclude:: /_verbatim/GMT_transverse_merc.txt

.. figure:: /_images/GMT_transverse_merc.*
   :width: 500 px
   :align: center

   Rectangular Transverse Mercator map.


A global transverse Mercator map - the equivalent of the 360° Mercator map - can also be obtained as follows:

.. literalinclude:: /_verbatim/GMT_TM.txt

.. _GMT_TM:

.. figure:: /_images/GMT_TM.*
   :width: 450 px
   :align: center

   A global transverse Mercator map.

Note that when a world map is given (indicated by **-R**\ *0/360/s/n*), the arguments are interpreted to mean oblique
degrees, i.e., the 360° range is understood to mean the extent of the plot along the central meridian, while the "south"
and "north" values represent how far from the central longitude we want the plot to extend. These values correspond to
latitudes in the regular Mercator projection and must therefore be less than 90.



.. _-Ju:

Universal Transverse Mercator (UTM) projection (**-Ju** **-JU**)
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

**Syntax**

    **-Ju**\|\ **U**\ *zone/*\ *scale*\|\ *width*

**Parameters**

- UTM *zone* (A, B, 1–60, Y, Z). Use negative values for numerical zones in the southern hemisphere or append the
  latitude modifiers (C–H, J–N, P–X) to specify an exact UTM grid zone.
- The *scale* along the equator in :ref:`plot-units <plt-units>`/degree or as 1:xxxxx (with **-Ju**) or map *width* in
  :ref:`plot-units <plt-units>` (with **-JU**).

**Description**

A particular subset of the :ref:`transverse Mercator <-Jt>` is the Universal Transverse Mercator (UTM) which was adopted
by the US Army for large-scale military maps. Here, the globe is divided into 60 zones between 84°S and 84°N, most of
which are 6° wide. Each of these UTM zones have a unique central meridian. Furthermore, each zone is divided into
latitude bands but these are not needed to specify the projection for most cases. See Figure
:ref:`Universal Transverse Mercator <GMT_utm_zones>` for all zone designations.

.. _GMT_utm_zones:

.. figure:: /_images/GMT_utm_zones.*
   :width: 700 px
   :align: center

   Universal Transverse Mercator zone layout.

.. toggle::

   Here is the source script for the figure above:

   .. literalinclude:: /_verbatim/GMT_utm_zones.txt

In order to minimize the distortion in any given zone, a scale factor of 0.9996 has been factored into the formulae
(although a standard, you can change this with :term:`PROJ_SCALE_FACTOR`). This makes the UTM projection a *secant*
projection and not a *tangent* projection like the :ref:`transverse Mercator <-Jt>` above. The scale only varies by 1
part in 1,000 from true scale at equator. The ellipsoidal projection expressions are accurate for map areas that extend
less than 10 away from the central meridian. For larger regions we use the conformal latitude in the general spherical
formulae instead.

.. _-Jo:

Oblique Mercator projection (**-Jo** **-JO**)
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

**Option 1 Syntax**

   **-Jo**\|\ **O**\ [**a**\|\ **A**]\ *lon0/lat0/azimuth/*\ *scale*\|\ *width*\ [**+v**]

**Option 1 Parameters**

   - The longitude (*lon0*) and latitude (*lat0*) of projection center.
   - The azimuth (*azimuth*) of the oblique equator.
   - The *scale* in :ref:`plot-units <plt-units>`/degree or 1:xxxxx along oblique equator (with **-Jo**),
     or map *width* in :ref:`plot-units <plt-units>` (with **-JO**).
   - Optionally, append **+v** to let the oblique Equator align with the *y*-axis [default is to align with the
     *x*-axis].

**Option 2 Syntax**

   **-Jo**\|\ **O**\ [**b**\|\ **B**]\ *lon0/lat0/lon1/lat1/*\ *scale*\|\ *width*\ [**+v**]

**Option 2 Parameters**

   - The longitude (*lon0*) and latitude (*lat0*) of projection center.
   - The longitude (*lon1*) and latitude (*lat1*) of a second point on oblique equator.
   - The *scale* in :ref:`plot-units <plt-units>`/degree or 1:xxxxx along oblique equator (with **-Jo**),
     or map *width* in :ref:`plot-units <plt-units>` (with **-JO**).
   - Optionally, append **+v** to let the oblique Equator align with the *y*-axis [default is to align with the
     *x*-axis].

**Option 3 Syntax**

   **-Jo**\|\ **O**\ [**c**\|\ **C**]\ *lon0/lat0/lonp/latp/*\ *scale*\|\ *width*\ [**+v**]

**Option 3 Parameters**

   - The longitude (*lon0*) and latitude (*lat0*) of projection center.
   - The longitude (*lonp*) and latitude (*latp*) of the projection pole.
   - The *scale* in :ref:`plot-units <plt-units>`/degree or 1:xxxxx along oblique equator (with **-Jo**),
     or map *width* in :ref:`plot-units <plt-units>` (with **-JO**).
   - Optionally, append **+v** to let the oblique Equator align with the *y*-axis [default is to align with the
     *x*-axis].

For all three definitions, the upper case **A**\|\ **B**\|\ **C** means we will allow projection poles in the southern
hemisphere [default is to map any such poles to their antipodes in the northern hemisphere].

**Description**

Oblique configurations of the cylinder give rise to the oblique Mercator projection. It is particularly useful when
mapping regions of large lateral extent in an oblique direction. Both parallels and meridians are complex curves. The
projection was developed in the early 1900s by several workers.

**Example**

An oblique view of some Caribbean islands using Option 3 can be obtained as follows:

.. literalinclude:: /_verbatim/GMT_obl_merc.txt

.. figure:: /_images/GMT_obl_merc.*
   :width: 500 px
   :align: center

   Oblique Mercator map using **-JOc**. We make it clear which direction is North by adding a star rose with the **-Td**
   option.


Note that we define our region using the rectangular system described earlier. If we do not append **+r** to the **-R**
string then the information provided with the **-R** option is assumed to be oblique degrees about the projection center
rather than the usual geographic coordinates. This interpretation is chosen since in general the parallels and meridians
are not very suitable as map boundaries.

When working with oblique projections such as here, it is often much more convenient to specify the map domain in the
projected coordinates relative to the map center. The figure below shows two views of New Zealand using the oblique
Mercator projection that in both cases specifies the region using **-R**\ -1000/1000/-500/500\ **+uk**.  The unit **k**
means the following bounds are in projected km and we let GMT determine the geographic coordinates of the two diagonal
corners internally.

.. figure:: /_images/GMT_obl_nz.*
   :width: 600 px
   :align: center

   (left) Oblique view of New Zealand centered on its geographical center (Nelson) indicated by the white circle for an
   oblique Equator with azimuth 35.  This resulted in the argument **-JOa**\ 173:17:02E/41:16:15S/35/3i. The map is
   2000 km by 1000 km and the Cartesian coordinate system in the projected units are indicated by the bold axes.  The
   blue circle is the point (40S,180E) and it has projected coordinates (*x* = 426.2, *y* = -399.7).
   (right) Same dimensions but now specifying an azimuth of 215, yielding a projection pole in the southern hemisphere
   (hence we used **-JOA** to override the restriction, i.e., **-JOA**\ 173:17:02E/41:16:15S/215/3i.)
   The projected coordinate system is still aligned as before but the Earth has been rotated 180 degrees.  The blue
   point now has projected coordinates (*x* = -426.2, *y* = 399.7).

.. toggle::

   Here is the source script for the figure above:

   .. literalinclude:: /_verbatim/GMT_obl_nz.txt

The oblique Mercator projection will by default arrange the output so that the oblique Equator becomes the new
horizontal, positive *x*-axis.  For features with an orientation more north-south than east-west, it may be preferable
to align the oblique Equator with the vertical, positive *y*-axis instead.  This configuration is selected by appending
**+v** to the **-J** projection option.  The example below shows this behaviour.

.. figure:: /_images/GMT_obl_baja.*
   :width: 300 px
   :align: center

   Oblique view of Baja California using the vertical oblique Equator modifier.  This plot
   resulted from the argument **-JOa**\ 120W/25N/-30/6c\ **+v**\ .

.. toggle::

   Here is the source script for the figure above:

   .. literalinclude:: /_verbatim/GMT_obl_baja.txt

.. _-Jc:

Cassini cylindrical projection (**-Jc** **-JC**)
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

**Syntax**

    **-Jc**\|\ **C**\ *lon0/lat0/scale*\|\ *width*

**Parameters**

   - The longitude (*lon0*) and latitude (*lat0*) of the central point.
   - The *scale* in :ref:`plot-units <plt-units>`/degree or as 1:xxxxx (with **-Jc**) or map *width* in
     :ref:`plot-units <plt-units>` (with **-JC**).

**Description**

This cylindrical projection was developed in 1745 by César-François Cassini de Thury for the survey of France. It is
occasionally called Cassini-Soldner since the latter provided the more accurate mathematical analysis that led to the
development of the ellipsoidal formulae. The projection is neither conformal nor equal-area, and behaves as a compromise
between the two end-members. The distortion is zero along the central meridian. It is best suited for mapping regions of
north-south extent. The central meridian, each meridian 90° away, and equator are straight lines; all other meridians
and parallels are complex curves.

**Example**

A detailed map of the island of Sardinia centered on the 8°45'E meridian using the Cassini projection can be obtained by
as follows:

.. literalinclude:: /_verbatim/GMT_cassini.txt

.. figure:: /_images/GMT_cassini.*
   :width: 400 px
   :align: center

   Cassini map over Sardinia.


As with the previous projections, the user can choose between a rectangular boundary (used here) or a geographical
(WESN) boundary.

.. _-Jq:

Cylindrical equidistant projection (**-Jq** **-JQ**)
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

**Syntax**

   **-Jq**\|\ **Q**\ [*lon0/*\ [*lat0/*]]\ *scale*\|\ *width*

**Parameters**

- Optionally, the central meridian (*lon0*) [default is the middle of the map map].
- Optionally, the standard parallel (*lat0*)  [default is the equator]. When supplied, the central meridian (*lon0*)
  must be supplied as well.
- The *scale*  in :ref:`plot-units <plt-units>`/degree or as 1:xxxxx (with **-Jq**) or map *width* in
  :ref:`plot-units <plt-units>` (with **-JQ**).

**Description**

This simple cylindrical projection is really a linear scaling of longitudes and latitudes. The most common form is the
Plate Carrée projection, where the scaling of longitudes and latitudes is the same. All meridians and parallels are
straight lines.

Different relative scalings of longitudes and latitudes can be obtained by selecting a standard parallel different from
the equator. Some selections for standard parallels have practical properties as shown in Table :ref:`JQ <tbl-JQ>`.

.. _tbl-JQ:

+-----------------------------------------------------+--------+
+=====================================================+========+
| Grafarend and Niermann, minimum linear distortion   | 61.7°  |
+-----------------------------------------------------+--------+
| Ronald Miller Equirectangular                       | 50.5°  |
+-----------------------------------------------------+--------+
| Ronald Miller, minimum continental distortion       | 43.5°  |
+-----------------------------------------------------+--------+
| Grafarend and Niermann                              | 42°    |
+-----------------------------------------------------+--------+
| Ronald Miller, minimum overall distortion           | 37.5°  |
+-----------------------------------------------------+--------+
| Plate Carrée, Simple Cylindrical, Plain/Plane       | 0°     |
+-----------------------------------------------------+--------+

**Example**

A world map centered on the dateline using this projection can be obtained as follows:

.. literalinclude:: /_verbatim/GMT_equi_cyl.txt

.. figure:: /_images/GMT_equi_cyl.*
   :width: 500 px
   :align: center

   World map using the Plate Carrée projection.

.. _-Jy:

Cylindrical equal-area projections (**-Jy** **-JY**)
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

**Syntax**

    **-Jy**\|\ **Y**\ [*lon0/*\ [*lat0/*]]\ *scale*\|\ *width*

**Parameters**

- Optionally, the central meridian (*lon0*) [default is the middle of the map].
- Optionally, the standard parallel (*lat0*) [default is the equator]. When supplied, the central meridian (*lon0*)
  must be supplied as well.
- The *scale* in :ref:`plot-units <plt-units>`/degree or as 1:xxxxx (with **-Jy**) or map *width* in
  :ref:`plot-units <plt-units>` (with **-JY**)

**Description**

This cylindrical projection is actually several projections, depending on what latitude is selected as the standard
parallel. However, they are all equal area and hence non-conformal. All meridians and parallels are straight lines.

While you may choose any value for the standard parallel and obtain your own personal projection, there are seven
choices of standard parallels that result in known (or named) projections. These are listed in Table :ref:`JY <tbl-JY>`.

.. _tbl-JY:

+-------------------+---------------------+
+===================+=====================+
| Balthasart        | 50°                 |
+-------------------+---------------------+
| Gall              | 45°                 |
+-------------------+---------------------+
| Hobo-Dyer         | 37°30' (= 37.5°)    |
+-------------------+---------------------+
| Trystan Edwards   | 37°24' (= 37.4°)    |
+-------------------+---------------------+
| Caster            | 37°04' (= 37.0666°) |
+-------------------+---------------------+
| Behrman           | 30°                 |
+-------------------+---------------------+
| Lambert           | 0°                  |
+-------------------+---------------------+

**Example**

A world map centered on the 35°E meridian using the Behrman projection (Figure
:ref:`Behrman cylindrical projection <GMT_general_cyl>`) can be obtained as follows:

.. literalinclude:: /_verbatim/GMT_general_cyl.txt

.. _GMT_general_cyl:

.. figure:: /_images/GMT_general_cyl.*
   :width: 600 px
   :align: center

   World map using the Behrman cylindrical equal-area projection.


As one can see there is considerable distortion at high latitudes since the poles map into lines.

.. _-Jj:

Miller Cylindrical projection (**-Jj** **-JJ**)
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

**Syntax**

    **-Jj**\|\ **J**\ [*lon0/*]\ *scale*\|\ *width*

**Parameters**

- Optionally, the central meridian (*lon0*) [default is the middle of the map].

- The *scale* in :ref:`plot-units <plt-units>`/degree or as 1:xxxxx (with **-Jj**) or map *width* in
  :ref:`plot-units <plt-units>` (with **-JJ**).

**Description**

This cylindrical projection, presented by Osborn Maitland Miller of the American Geographic Society in 1942, is neither
equal nor conformal. All meridians and parallels are straight lines. The projection was designed to be a compromise
between Mercator and other cylindrical projections. Specifically, Miller spaced the parallels by using Mercator's
formula with 0.8 times the actual latitude, thus avoiding the singular poles; the result was then divided by 0.8. There
is only a spherical form for this projection.

**Example**

A world map centered on the 90°E meridian at a map scale of 1:400,000,000 (Figure :ref:`Miller projection <GMT_miller>`)
can be obtained as follows:

.. literalinclude:: /_verbatim/GMT_miller.txt

.. _GMT_miller:

.. figure:: /_images/GMT_miller.*
   :width: 500 px
   :align: center

   World map using the Miller cylindrical projection.

.. _-Jcyl_stere:

Cylindrical stereographic projections (**-Jcyl_stere** **-JCyl_stere**)
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

**Syntax**

    **-Jcyl_stere**\|\ **Cyl_stere**\ /[*lon0/*\ [*lat0/*]]\ *scale*\|\ *width*

**Parameters**

- Optionally, the central meridian (*lon0*) [default is the middle of the map].
- Optionally, the standard parallel (*lat0*) [default is the Equator]. When used, central meridian (*lon0*) needs to be
  given as well.
- The *scale* in :ref:`plot-units <plt-units>`/degree or as 1:xxxxx (with **-Jcyl_stere**) or map *width* in
  :ref:`plot-units <plt-units>` (with **-JCyl_stere**).

**Description**

The cylindrical stereographic projections are certainly not as notable as other cylindrical projections, but are still
used because of their relative simplicity and their ability to overcome some of the downsides of other cylindrical
projections, like extreme distortions of the higher latitudes. The stereographic projections are perspective
projections, projecting the sphere onto a cylinder in the direction of the antipodal point on the equator. The cylinder
crosses the sphere at two standard parallels, equidistant from the equator.

Some of the selections of the standard parallel are named for the cartographer or publication that popularized the
projection (Table :ref:`JCylstere <tbl-JCylstere>`).

.. _tbl-JCylstere:

+---------------------------------------------------------+-------------+
+=========================================================+=============+
| Miller's modified Gall                                  | 66.159467°  |
+---------------------------------------------------------+-------------+
| Kamenetskiy's First                                     | 55°         |
+---------------------------------------------------------+-------------+
| Gall's stereographic                                    | 45°         |
+---------------------------------------------------------+-------------+
| Bolshoi Sovietskii Atlas Mira or Kamenetskiy's Second   | 30°         |
+---------------------------------------------------------+-------------+
| Braun's cylindrical                                     | 0°          |
+---------------------------------------------------------+-------------+

**Example**

A map of the world, centered on the Greenwich meridian, using the Gall's stereographic projection (standard parallel is
45°, Figure :ref:`Gall's stereographic projection <GMT_gall_stereo>`), can be obtained as follows:

.. literalinclude:: /_verbatim/GMT_gall_stereo.txt

.. _GMT_gall_stereo:

.. figure:: /_images/GMT_gall_stereo.*
   :width: 500 px
   :align: center

   World map using Gall's stereographic projection.


Miscellaneous projections
-------------------------

GMT supports eight common projections for global presentation of data or models. These are the :ref:`Hammer <-Jh>`,
:ref:`Mollweide <-Jw>`, :ref:`Winkel Tripel <-Jr>`, :ref:`Robinson <-Jn>`, :ref:`Eckert IV and VI <-Jk>`,
:ref:`Sinusoidal <-Ji>`, and :ref:`Van der Grinten <-Jv>` projections. Due to the small scale used for global maps these
projections all use the spherical approximation rather than more elaborate elliptical formulae.

In all cases, the specification of the central meridian can be skipped. The default is the middle of the longitude
range of the plot, specified by the (**-R**) option.

.. _-Jh:

Hammer projection (**-Jh** **-JH**)
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

**Syntax**

    **-Jh**\|\ **H**\ [*lon0/*]\ *scale*\|\ *width*

**Parameters**

- The central meridian (*lon0*) [default is the middle of the map].
- The *scale* along equator in :ref:`plot-units <plt-units>`/degree or as 1:xxxxx (with **-Jh**) or map *width* in
  :ref:`plot-units <plt-units>` (with **-JH**).

**Description**

The equal-area Hammer projection, first presented by the German mathematician Ernst von Hammer in 1892, is also known as
Hammer-Aitoff (the Aitoff projection looks similar, but is not equal-area). The border is an ellipse, equator and
central meridian are straight lines, while other parallels and meridians are complex curves.

**Example**

A view of the Pacific ocean using the Dateline as central meridian can be generated thus:

.. literalinclude:: /_verbatim/GMT_hammer.txt

.. figure:: /_images/GMT_hammer.*
   :width: 500 px
   :align: center

   World map using the Hammer projection.

.. _-Jw:

Mollweide projection (**-Jw** **-JW**)
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

**Syntax**

    **-Jw**\|\ **W**\ [*lon0/*]\ *scale*\|\ *width*

**Parameters**

- The central meridian (*lon0*) [default is the middle of the map].
- The *scale* along equator in :ref:`plot-units <plt-units>`/degree or as 1:xxxxx (with **-Jw**) or map *width* in
  :ref:`plot-units <plt-units>` (with **-JW**).

**Description**

This pseudo-cylindrical, equal-area projection was developed by the German mathematician and astronomer Karl Brandan
Mollweide in 1805. Parallels are unequally spaced straight lines with the meridians being equally spaced elliptical
arcs. The scale is only true along latitudes 40°44' north and south. The projection is used mainly for global maps
showing data distributions. It is occasionally referenced under the name *homalographic* projection.

**Example**

An example centered on Greenwich can be generated thus:

.. literalinclude:: /_verbatim/GMT_mollweide.txt

.. figure:: /_images/GMT_mollweide.*
   :width: 500 px
   :align: center

   World map using the Mollweide projection.

.. _-Jr:

Winkel Tripel projection (**-Jr** **-JR**)
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

**Syntax**

    **-Jr**\|\ **R**\ [*lon0/*]\ *scale*\|\ *width*

**Parameters**

- The central meridian (*lon0*) [default is the middle of the map].
- The *scale* along equator in :ref:`plot-units <plt-units>`/degree or as 1:xxxxx (with **-Jr**) or map *width* in
  :ref:`plot-units <plt-units>` (with **-JR**).

**Description**

In 1921, the German mathematician Oswald Winkel created a projection that was to strike a compromise between the
properties of three elements (area, angle and distance). The German word "tripel" refers to this junction of where each
of these elements are least distorted when plotting global maps. The projection was popularized when Bartholomew and Son
started to use it in its world-renowned "The Times Atlas of the World" in the mid-20th century. In 1998, the National
Geographic Society made the Winkel Tripel as its map projection of choice for global maps.

Naturally, this projection is neither conformal, nor equal-area. Central meridian and equator are straight lines; other
parallels and meridians are curved. The projection is obtained by averaging the coordinates of the Equidistant
Cylindrical and Aitoff (not Hammer-Aitoff) projections. The poles map into straight lines 0.4 times the length of
equator.

**Example**

Centered on Greenwich, the example in Figure :ref:`Winkel Tripel projection <GMT_winkel>` was created by this command:

.. literalinclude:: /_verbatim/GMT_winkel.txt

.. _GMT_winkel:

.. figure:: /_images/GMT_winkel.*
   :width: 500 px
   :align: center

   World map using the Winkel Tripel projection.

.. _-Jn:

Robinson projection (**-Jn** **-JN**)
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

**Syntax**

    **-Jn**\|\ **N**\ [*lon0/*]\ *scale*\|\ *width*

**Parameters**

- The central meridian (*lon0*) [default is the middle of the map].
- The *scale* along equator in :ref:`plot-units <plt-units>`/degree or as 1:xxxxx (with **-Jn**) or map *width* in
  :ref:`plot-units <plt-units>` (with **-JN**).

**Description**

The Robinson projection, presented by the American geographer and cartographer Arthur H. Robinson in 1963, is a modified
cylindrical projection that is neither conformal nor equal-area. Central meridian and all parallels are straight lines;
other meridians are curved. It uses lookup tables rather than analytic expressions to make the world map "look"
right\ [22]_. The scale is true along latitudes 38. The projection was originally developed for use by Rand McNally and
is currently used by the National Geographic Society.

**Example**

Again centered on Greenwich, the example below was created by this command:

.. literalinclude:: /_verbatim/GMT_robinson.txt

.. figure:: /_images/GMT_robinson.*
   :width: 500 px
   :align: center

   World map using the Robinson projection.

.. _-Jk:

Eckert IV and VI projection (**-Jk** **-JK**)
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

**Syntax**

   **-Jk**\|\ **K**\ **f**\ [*lon0/*]\ *scale*\|\ *width* (Eckert IV)
   **-Jk**\|\ **K**\ [**s**][*lon0/*]\ *scale*\|\ *width* (Eckert VI)

**Parameters**

- The central meridian (*lon0*) [default is the middle of the map].
- The *scale* along equator in :ref:`plot-units <plt-units>`/degree or as 1:xxxxx (with **-Jk**) or map *width* in
  :ref:`plot-units <plt-units>` (with **-JK**).

**Description**

The Eckert IV and VI projections, presented by the German cartographer Max Eckert-Greiffendorff in 1906, are
pseudo-cylindrical equal-area projections. Central meridian and all parallels are straight lines; other meridians are
equally spaced elliptical arcs (IV) or sinusoids (VI). The scale is true along latitudes 40°30' (IV) and 49°16' (VI).
Their main use is in thematic world maps. To select Eckert IV you must use **-JKf** (**f** for "four") while Eckert VI
is selected with **-JKs** (**s** for "six"). If no modifier is given it defaults to Eckert VI.

**Examples**

Centered on the Dateline, the Eckert IV example below was created by this command:

.. literalinclude:: /_verbatim/GMT_eckert4.txt

.. figure:: /_images/GMT_eckert4.*
   :width: 500 px
   :align: center

   World map using the Eckert IV projection.


The same script, with **s** instead of **f**, yields the Eckert VI map:

.. figure:: /_images/GMT_eckert6.*
   :width: 500 px
   :align: center

   World map using the Eckert VI projection.

.. toggle::

   Here is the source script for the figure above:

   .. literalinclude:: /_verbatim/GMT_eckert6.txt

.. _-Ji:

Sinusoidal projection (**-Ji** **-JI**)
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

**Syntax**

    **-Ji**\|\ **I**\ [*lon0/*]\ *scale*\|\ *width*

**Parameters**

- The central meridian (*lon0*) [default is the middle of the map].
- The *scale* along equator in :ref:`plot-units <plt-units>`/degree or as 1:xxxxx (with **-Ji**) or map *width* in
  :ref:`plot-units <plt-units>` (with **-JI**).

**Description**

The sinusoidal projection is one of the oldest known projections, is equal-area, and has been used since the mid-16th
century. It has also been called the "Equal-area Mercator" projection. The central meridian is a straight line; all
other meridians are sinusoidal curves. Parallels are all equally spaced straight lines, with scale being true along all
parallels (and central meridian).

**Examples**

A simple world map using the sinusoidal projection is therefore obtained by

.. literalinclude:: /_verbatim/GMT_sinusoidal.txt

.. figure:: /_images/GMT_sinusoidal.*
   :width: 500 px
   :align: center

   World map using the Sinusoidal projection.


To reduce distortion of shape the interrupted sinusoidal projection was introduced in 1927. Here, three symmetrical
segments are used to cover the entire world. Traditionally, the interruptions are at 160°W, 20°W, and 60°E. To make the
interrupted map we must call :doc:`/coast` for each segment and superpose the results. To produce an interrupted world
map (with the traditional boundaries just mentioned) that is 14.4 cm wide we use the scale 14.4/360 = 0.04 and offset
the subsequent plots horizontally by their widths (140\ :math:`\cdot`\ 0.04 and 80\ :math:`\cdot`\ 0.04):

.. literalinclude:: /_verbatim/GMT_sinus_int.txt

.. figure:: /_images/GMT_sinus_int.*
   :width: 500 px
   :align: center

   World map using the Interrupted Sinusoidal projection.


The usefulness of the interrupted sinusoidal projection is basically limited to display of global, discontinuous data
distributions like hydrocarbon and mineral resources, etc.

.. _-Jv:

Van der Grinten projection (**-Jv** **-JV**)
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

**Syntax**

    **-Jv**\|\ **V**\ [*lon0/*]\ *scale*\|\ *width*

**Parameters**

- The central meridian (*lon0*) [default is the middle of the map].
- The *scale* along equator in :ref:`plot-units <plt-units>`/degree or as 1:xxxxx (with **-Jv**) or map *width* in
  :ref:`plot-units <plt-units>` (with **-JV**).

**Description**

The Van der Grinten projection, presented by Alphons J. van der Grinten in 1904, is neither equal-area nor conformal.
Central meridian and Equator are straight lines; other meridians are arcs of circles. The scale is true along the
Equator only. Its main use is to show the entire world enclosed in a circle.

**Example**

Centered on the Dateline, the example below was created by this command:

.. literalinclude:: /_verbatim/GMT_grinten.txt

.. figure:: /_images/GMT_grinten.*
   :width: 400 px
   :align: center

   World map using the Van der Grinten projection.

Footnotes
---------

.. [20]
   Snyder, J. P., 1987, Map Projections A Working Manual, U.S.
   Geological Survey Prof. Paper 1395.

.. [21]
   This is, however, not the shortest distance. It is given by the great
   circle connecting the two points.

.. [22]
   Robinson provided a table of *y*-coordinates for latitudes
   every 5. To project values for intermediate latitudes one must
   interpolate the table. Different interpolants may result in slightly
   different maps. GMT uses the
   interpolant selected by the parameter :term:`GMT_INTERPOLANT` in the
   file.
