1GMT Map Projections 2=================== 3 4GMT implements more than 30 different projections. They all project the input coordinates longitude and latitude to 5positions 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 6the radial vector length to the :math:`(x,y)` point on the chosen ellipsoid. The functions :math:`f` and :math:`g` can be 7quite nasty and we will refrain from presenting details in this document. The interested reader is referred to *Snyder* 8[1987]\ [20]_. We will mostly be using the :doc:`/coast` command to demonstrate each of the projections. GMT map 9projections are grouped into four categories depending on the nature of the projection. The groups are 10 11#. :ref:`cookbook/map-projections:Conic projections` 12 13#. :ref:`cookbook/map-projections:Azimuthal projections` 14 15#. :ref:`cookbook/map-projections:Cylindrical projections` 16 17#. :ref:`cookbook/map-projections:Miscellaneous projections` 18 19Because :math:`x` and :math:`y` are coupled we can only specify one plot-dimensional scale, typically a map *scale* 20(for lower-case map projection code) or a map *width* (for upper-case map projection code). The *measurement unit* 21is cm, inch, or point, depending on the :term:`PROJ_LENGTH_UNIT` setting in **gmt.conf**, but this can be overridden on 22the command line by appending **c**, **i**, or **p** to the *scale* or *width* values. In some cases it would be more 23practical to specify map *height* instead of *width*, while in other situations it would be nice to set either the 24*shortest* or *longest* map dimension. Users may select these alternatives by appending a character code to their map 25dimension [detault is **+dw**]: 26 27 - Append **+dh** to the given :ref:`dimension <cookbook/features:Dimension units>` to specify map *height*. 28 - Append **+du** to the given :ref:`dimension <cookbook/features:Dimension units>` to select the minimum map 29 dimension. 30 - Append **+dl** to the given :ref:`dimension <cookbook/features:Dimension units>` to select the maximum map 31 dimension. 32 33The ellipsoid used in map projections is user-definable. 73 commonly used ellipsoids and spheroids are currently 34supported, and users may also specify their own custom ellipsoid parameters [default is WGS-84]. Several GMT parameters 35can affect the projection: :term:`PROJ_ELLIPSOID`, :term:`GMT_INTERPOLANT`, :term:`PROJ_SCALE_FACTOR`, and 36:term:`PROJ_LENGTH_UNIT`; see the :doc:`../gmt.conf` man page for details. 37 38In GMT version 4.3.0 we noticed we ran out of the alphabet for 1-letter (and sometimes 2-letter) projection codes. To 39allow more flexibility, and to make it easier to remember the codes, we implemented the option to use the abbreviations 40used by the `PROJ <https://proj.org/>`_ mapping package. Since some of the GMT projections are not in **PROJ**, we 41invented some of our own as well. For a full list of both the old 1- and 2-letter codes, as well as the 42**PROJ**-equivalents see the quick reference table below. For example, **-JM**\ 15c and **-JMerc**\ /15c have the same 43meaning. 44 45.. include:: ../proj-codes.rst_ 46 47Conic projections 48----------------- 49 50.. _-Jb: 51 52Albers conic equal-area projection (**-Jb** **-JB**) 53~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ 54 55**Syntax** 56 57 **-Jb**\|\ **B**\ *lon0/lat0/lat1/lat2/*\ *scale*\|\ *width* 58 59**Parameters** 60 61- The longitude (*lon0*) and latitude (*lat0*) of the projection center. 62- The two standard parallels (*lat1* and *lat2*). 63- The *scale* in :ref:`plot-units <plt-units>`/degree or as 1:xxxxx (with **-Jb**) or map *width* in 64 :ref:`plot-units <plt-units>` (with **-JB**). 65 66Note that you must include the "1:" if you choose to specify the *scale* that way. For example, you can say 0.5c which 67means 0.5 cm/degree or 1:200000 which means 1 unit on the map equals 200,000 units along the standard parallels. The 68projection center defines the origin of the rectangular map coordinates. 69 70**Description** 71 72This projection, developed by Heinrich C. Albers in 1805, is predominantly used to map regions of large east-west 73extent, in particular the United States. It is a conic, equal-area projection, in which parallels are unequally 74spaced arcs of concentric circles, more closely spaced at the north and south edges of the map. Meridians, on the other 75hand, are equally spaced radii about a common center, and cut the parallels at right angles. Distortion in scale and 76shape vanishes along the two standard parallels. Between them, the scale along parallels is too small; beyond them it is 77too large. The opposite is true for the scale along meridians. 78 79**Example** 80 81As 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 82and 25°N and 45°N as our two standard parallels. We desire a map that is 12 cm wide. The complete command needed to 83generate the map below is therefore given by: 84 85.. literalinclude:: /_verbatim/GMT_albers.txt 86 87.. figure:: /_images/GMT_albers.* 88 :width: 500 px 89 :align: center 90 91 Albers equal-area conic map projection. 92 93.. _-Jd: 94 95Equidistant conic projection (**-Jd** **-JD**) 96~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ 97 98**Syntax** 99 100 **-Jd**\|\ **D**\ *lon0/lat0/lat1/lat2/*\ *scale*\|\ *width* 101 102**Parameters** 103 104- The longitude (*lon0*) and latitude (*lat0*) of the projection center. 105- The two standard parallels (*lat1* and *lat2*). 106- The *scale* in :ref:`plot-units <plt-units>`/degree or as 1:xxxxx (with **-Jd**) or map *width* in 107 :ref:`plot-units <plt-units>` (with **-JD**). 108 109**Description** 110 111The equidistant conic projection was described by the Greek philosopher Claudius Ptolemy about A.D. 150. It is neither 112conformal or equal-area, but serves as a compromise between them. The scale is true along all meridians and the standard 113parallels. 114 115**Example** 116 117The equidistant conic projection is often used for atlases with maps of small countries. As an example, we generate a 118map of Cuba: 119 120.. literalinclude:: /_verbatim/GMT_equidistant_conic.txt 121 122.. figure:: /_images/GMT_equidistant_conic.* 123 :width: 500 px 124 :align: center 125 126 Equidistant conic map projection. 127 128.. _-Jl: 129 130Lambert conic conformal projection (**-Jl** **-JL**) 131~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ 132 133**Syntax** 134 135 **-Jl**\|\ **L**\ *lon0/lat0/lat1/lat2/*\ *scale*\|\ *width* 136 137**Parameters** 138 139- The longitude (*lon0*) and latitude (*lat0*) of the projection center. 140- The two standard parallels (*lat1* and *lat2*). 141- The *scale* in :ref:`plot-units <plt-units>`/degree or as 1:xxxxx (with **-Jl**) or map *width* in 142 :ref:`plot-units <plt-units>` (with **-JL**). 143 144**Description** 145 146This conic projection was designed by the Alsatian mathematician Johann Heinrich Lambert (1772) and has been used 147extensively for mapping of regions with predominantly east-west orientation, just like the Albers projection. Unlike the 148Albers projection, Lambert's conformal projection is not equal-area. The parallels are arcs of circles with a common 149origin, and meridians are the equally spaced radii of these circles. As with Albers projection, it is only the two 150standard parallels that are distortion-free. 151 152**Example** 153 154The Lambert conformal projection has been used for basemaps for all the 48 contiguous States with the two fixed standard 155parallels 33°N and 45°N. We will generate a map of the continental USA using these parameters. Note that with all the 156projections you have the option of selecting a rectangular border rather than one defined by meridians and parallels. 157Here, we choose the regular WESN region, a "fancy" basemap frame, and use degrees west for longitudes. The generating 158commands used were: 159 160.. literalinclude:: /_verbatim/GMT_lambert_conic.txt 161 162.. figure:: /_images/GMT_lambert_conic.* 163 :width: 500 px 164 :align: center 165 166 Lambert conformal conic map projection. 167 168 169The choice for projection center does not affect the projection but it indicates which meridian (here 100°W) will be 170vertical on the map. The standard parallels were originally selected by Adams to provide a maximum scale error between 171latitudes 30.5°N and 47.5°N of 0.5–1%. Some areas, like Florida, experience scale errors of up to 2.5%. 172 173.. _-Jpoly: 174 175(American) polyconic projection (**-Jpoly** **-JPoly**) 176~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ 177 178**Syntax** 179 180 **-Jpoly**\|\ **Poly**\ /[*lon0/*\ [*lat0/*]]\ *scale*\|\ *width* 181 182**Parameters** 183 184- The longitude (*lon0*) and latitude (*lat0*) of the projection center. 185- The two standard parallels (*lat1* and *lat2*). 186- The *scale* in :ref:`plot-units <plt-units>`/degree or as 1:xxxxx (with **-Jl**) or map *width* in 187 :ref:`plot-units <plt-units>` (with **-JL**). 188 189**Description** 190 191The polyconic projection, in Europe usually referred to as the American polyconic projection, was introduced shortly 192before 1820 by the Swiss-American cartographer Ferdinand Rodulph Hassler (1770–1843). As head of the Survey of the 193Coast, he was looking for a projection that would give the least distortion for mapping the coast of the United 194States. The projection acquired its name from the construction of each parallel, which is achieved by projecting the 195parallel onto the cone while it is rolled around the globe, along the central meridian, tangent to that parallel. As a 196consequence, the projection involves many cones rather than a single one used in regular conic projections. 197 198The polyconic projection is neither equal-area, nor conformal. It is true to scale without distortion along the central 199meridian. Each parallel is true to scale as well, but the meridians are not as they get further away from the central 200meridian. As a consequence, no parallel is standard because conformity is lost with the lengthening of the meridians. 201 202**Example** 203 204Below we reproduce the illustration by *Snyder* [1987], with a gridline every 10 and annotations only every 30° in 205longitude: 206 207.. literalinclude:: /_verbatim/GMT_polyconic.txt 208 209.. figure:: /_images/GMT_polyconic.* 210 :width: 500 px 211 :align: center 212 213 (American) polyconic projection. 214 215 216Azimuthal projections 217--------------------- 218 219.. _-Ja: 220 221Lambert Azimuthal Equal-Area (**-Ja** **-JA**) 222~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ 223 224**Syntax** 225 226 **-Ja**\|\ **A**\ *lon0/lat0*\ [*/horizon*]\ *scale*\|\ *width* 227 228**Parameters** 229 230- The longitude (*lon0*) and latitude (*lat0*) of the projection center. 231- Optionally, the *horizon*, i.e., the number of degrees from the center to the edge (<=180) [default is 90]. 232- The *scale* as 1:xxxxx or as radius/latitude where radius is the projected distance on the map from projection center 233 to an oblique latitude where 0 would be the oblique Equator (with **-Ja**) or map *width* 234 :ref:`plot-units <plt-units>` (with **-JA**). 235 236**Description** 237 238This projection was developed by Johann Heinrich Lambert in 1772 and is typically used for mapping large regions like 239continents and hemispheres. It is an azimuthal, equal-area projection, but is not perspective. Distortion is zero at the 240center of the projection, and increases radially away from this point. 241 242**Examples** 243 244Two different types of maps can be made with this projection depending on how the region is specified. We will give 245examples of both types in the next two subsections. 246 247Rectangular map 248^^^^^^^^^^^^^^^ 249 250In this mode we define our region by specifying the longitude/latitude 251of the lower left and upper right corners instead of the usual *west, 252east, south, north* boundaries. The reason for specifying our area this 253way is that for this and many other projections, lines of equal 254longitude and latitude are not straight lines and are thus poor choices 255for map boundaries. Instead we require that the map boundaries be 256rectangular by defining the corners of a rectangular map boundary. Using 2570°E/40°S (lower left) and 60°E/10°S (upper right) as our corners we try 258 259.. literalinclude:: /_verbatim/GMT_lambert_az_rect.txt 260 261.. figure:: /_images/GMT_lambert_az_rect.* 262 :width: 500 px 263 :align: center 264 265 Rectangular map using the Lambert azimuthal equal-area projection. 266 267 268Note that an **+r** is appended to the **-R** option to inform GMT that 269the region has been selected using the rectangle technique, otherwise it 270would try to decode the values as *west, east, south, north* and report 271an error since *'east'* < *'west'*. 272 273Hemisphere map 274^^^^^^^^^^^^^^ 275 276Here, you must specify the world as your region (**-Rg** or **-Rd**). E.g., to obtain a hemisphere view that shows the 277Americas, try 278 279.. literalinclude:: /_verbatim/GMT_lambert_az_hemi.txt 280 281.. figure:: /_images/GMT_lambert_az_hemi.* 282 :width: 400 px 283 :align: center 284 285 Hemisphere map using the Lambert azimuthal equal-area projection. 286 287 288To geologists, the Lambert azimuthal equal-area projection (with origin 289at 0/0) is known as the *equal-area* (Schmidt) stereonet and used for 290plotting fold axes, fault planes, and the like. An *equal-angle* (Wulff) 291stereonet can be obtained by using the stereographic projection 292(discussed later). The stereonets produced by these two projections appear below. 293 294.. _GMT_stereonets: 295 296.. figure:: /_images/GMT_stereonets.* 297 :width: 500 px 298 :align: center 299 300 Equal-Area (Schmidt) and Equal-Angle (Wulff) stereo nets. 301 302.. toggle:: 303 304 Here is the source script for the figure above: 305 306 .. literalinclude:: /_verbatim/GMT_stereonets.txt 307 308 309.. _-Js: 310 311Stereographic Equal-Angle (**-Js** **-JS**) 312~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ 313 314**Syntax** 315 316 **-Js**\|\ **S**\ *lon0/lat0*\ [*/horizon*]\ */*\ *scale*\|\ *width* 317 318**Parameters** 319 320- The longitude (*lon0*) and latitude (*lat0*) of the projection center. 321- Optionally, the *horizon*, i.e., the number of degrees from the center to the edge (< 180) [default is 90]. 322- Scale as 1:xxxxx (true scale at pole), slat/1:xxxxx (true scale at standard parallel slat), or radius/latitude where 323 radius is distance on map in :ref:`plot-units <plt-units>` from projection center to a particular oblique latitude 324 (with **-Js**) or simply map *width* in :ref:`plot-units <plt-units>` (with **-JS**). 325 326**Description** 327 328 329This is a conformal, azimuthal projection that dates back to the Greeks. Its main use is for mapping the polar regions. 330In the polar aspect all meridians are straight lines and parallels are arcs of circles. While this is the most common 331use it is possible to select any point as the center of projection. The requirements are 332 333A map scale factor of 0.9996 will be applied by default (although you may change this with :term:`PROJ_SCALE_FACTOR`). 334However, the setting is ignored when a standard parallel has been specified since the scale is then implicitly given. 335We will look at two different types of maps. 336 337**Examples** 338 339Multiple types of maps can be made with this projection depending on how the region is specified. We will give 340examples in the next three subsections. 341 342Polar Stereographic Map 343^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ 344 345In our first example we will let the projection center be at the north 346pole. This means we have a polar stereographic projection and the map 347boundaries will coincide with lines of constant longitude and latitude. 348An example is given by: 349 350.. literalinclude:: /_verbatim/GMT_stereographic_polar.txt 351 352.. figure:: /_images/GMT_stereographic_polar.* 353 :width: 500 px 354 :align: center 355 356 Polar stereographic conformal projection. 357 358 359Rectangular stereographic map 360^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ 361 362As with Lambert's azimuthal equal-area projection we have the option to 363use rectangular boundaries rather than the wedge-shape typically 364associated with polar projections. This choice is defined by selecting 365two points as corners in the rectangle and appending **+r** to the 366**-R** option. This command produces a map as presented in 367Figure :ref:`Polar stereographic <GMT_stereographic_rect>`: 368 369.. literalinclude:: /_verbatim/GMT_stereographic_rect.txt 370 371.. _GMT_stereographic_rect: 372 373.. figure:: /_images/GMT_stereographic_rect.* 374 :width: 500 px 375 :align: center 376 377 Polar stereographic conformal projection with rectangular borders. 378 379 380General stereographic map 381^^^^^^^^^^^^^^^^^^^^^^^^^ 382 383In terms of usage this projection is identical to the Lambert azimuthal 384equal-area projection. Thus, one can make both rectangular and 385hemispheric maps. Our example shows Australia using a projection pole at 386130°E/30°S. The command used was 387 388.. literalinclude:: /_verbatim/GMT_stereographic_general.txt 389 390.. figure:: /_images/GMT_stereographic_general.* 391 :width: 500 px 392 :align: center 393 394 General stereographic conformal projection with rectangular borders. 395 396 397By choosing 0/0 as the pole, we obtain the conformal stereonet presented 398next 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>`). 399 400.. _-Jg_pers: 401 402Perspective projection (**-Jg** **-JG**) 403~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ 404 405**Syntax** 406 407 **-Jg**\|\ **G**\ *lon0/lat0*\ */*\ *scale*\|\ *width*\ [**+a**\ *azimuth*][**+t**\ *tilt*][**+v**\ *vwidth/vheight*][**+w**\ *twist*][**+z**\ *altitude*\ [**r**\|\ **R**]\|\ **g**] 408 409**Required Parameters** 410 411- The longitude (*lon0*) and latitude (*lat0*) of the projection center. 412- The *scale* as 1:xxxxx or as radius/latitude where radius is distance on map in :ref:`plot-units <plt-units>` from 413 projection center to a particular oblique latitude (with **-Jg**), or map width in :ref:`plot-units <plt-units>` 414 (with **-JG**). 415 416**Optional Parameters** 417 418- The *azimuth* in degrees. This is the direction in which you are looking, measured clockwise from north [0]. 419- The *tilt* in degrees. This is the viewing angle relative to zenith. For example, a tilt of 0° is looking straight 420 down, and 60° is looking from 30° above the horizon [0]. 421- The *vwidth* and *vheight* of the viewpoint in degrees. This number depends on whether you are looking with the naked 422 eye (in which case the view is about 60° wide), or with binoculars, for example [unrestricted]. 423- The *twist* in degrees. This is the boresight rotation (clockwise) of the image [0]. 424- The *altitude* of the viewer above sea level in kilometers [infinity]. Alternatively, append **R** if giving 425 the distance of the viewer from the center of the Earth in Earth radii, or **r** if giving the distance from the 426 center of the Earth in kilometers. Finally, give *altitude* as **g** to compute and use the altitude for a 427 geosynchronous orbit. 428 429**Description** 430 431The perspective projection imitates in 2 dimensions the 3-dimensional view of the earth from space. The implementation 432in GMT is very flexible, and thus requires many input variables. 433 434**Example** 435 436The imagined view of northwest Europe from a Space Shuttle at 230 km looking due east is thus accomplished by the 437following :doc:`/coast` command (*lon0*\ =4; *lat0*\ =52; *altitude*\ =230 km; *azimuth*\ = 90°; *tilt*\ = 60°; 438*twist*\ = 180°; *vwidth*\ = *vheight*\ = 60°; and *width* = 12 cm): 439 440.. literalinclude:: /_verbatim/GMT_perspective.txt 441 442.. _GMT_perspective: 443 444.. figure:: /_images/GMT_perspective.* 445 :width: 500 px 446 :align: center 447 448 View from the Space Shuttle in Perspective projection. 449 450.. _-Jg: 451 452Orthographic projection (**-Jg** **-JG**) 453~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ 454 455**Syntax** 456 457 **-Jg**\|\ **G**\ *lon0/lat0*\ [*/horizon*]\ */*\ *scale*\|\ *width* 458 459**Parameters** 460 461- The longitude (*lon0*) and latitude (*lat0*) of the projection center. 462- Optionally, the *horizon*, i.e., the number of degrees from the center to the edge (<=90) [default is 90]. 463- The *scale* as 1:xxxxx or as radius/latitude where radius is distance on map in :ref:`plot-units <plt-units>` from 464 projection center to a particular oblique latitude (with **-Jg**), or map width in :ref:`plot-units <plt-units>` 465 (with **-JG**). 466 467**Description** 468 469The orthographic azimuthal projection is a perspective projection from infinite distance. It is therefore often used to 470give the appearance of a globe viewed from outer space. As with Lambert's equal-area and the stereographic projection, 471only one hemisphere can be viewed at any time. The projection is neither equal-area nor conformal, and much distortion 472is introduced near the edge of the hemisphere. The directions from the center of projection are true. The projection was 473known to the Egyptians and Greeks more than 2,000 years ago. Because it is mainly used for pictorial views at a small 474scale, only the spherical form is necessary. 475 476**Example** 477 478Our example of a perspective view centered on 75°W/40°N can therefore be generated by the following :doc:`/coast` 479command: 480 481.. literalinclude:: /_verbatim/GMT_orthographic.txt 482 483.. figure:: /_images/GMT_orthographic.* 484 :width: 400 px 485 :align: center 486 487 Hemisphere map using the Orthographic projection. 488 489.. _-Je: 490 491Azimuthal Equidistant projection (**-Je** **-JE**) 492~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ 493 494**Syntax** 495 496 **-Je**\|\ **E**\ *lon0/lat0*\ [*/horizon*]\ *scale*\|\ *width* 497 498**Parameters** 499 500- The longitude (*lon0*) and latitude (*lat0*) of the projection center. 501- Optionally, the *horizon*, i.e., the number of degrees from the center to the edge (<=180) [default is 180]. 502- The *scale* as 1:xxxxx or as radius/latitude where radius is distance on map in :ref:`plot-units <plt-units>` from 503 projection center to a particular oblique latitude (with **-Je**), or map width in :ref:`plot-units <plt-units>` 504 (with **-JE**). 505 506**Description** 507 508The most noticeable feature of this azimuthal projection is the fact that distances measured from the center are true. 509Therefore, a circle about the projection center defines the locus of points that are equally far away from the plot 510origin. Furthermore, directions from the center are also true. The projection, in the polar aspect, is at least several 511centuries old. It is a useful projection for a global view of locations at various or identical distance from a given 512point (the map center). 513 514**Example** 515 516Our example of a global view centered on 100°W/40°N can therefore be generated by the following :doc:`/coast` command. 517Note that the antipodal point is 180° away from the center, but in this projection this point plots as the entire map 518perimeter: 519 520.. literalinclude:: /_verbatim/GMT_az_equidistant.txt 521 522.. figure:: /_images/GMT_az_equidistant.* 523 :width: 400 px 524 :align: center 525 526 World map using the equidistant azimuthal projection. 527 528.. _-Jf: 529 530Gnomonic projection (**-Jf** **-JF**) 531~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ 532 533**Syntax** 534 535 **-Jf**\|\ **F**\ *lon0/lat0*\ [*/horizon*]\ *scale*\|\ *width* 536 537**Parameters** 538 539- The longitude (*lon0*) and latitude (*lat0*) of the projection center. 540- Optionally, the *horizon*, i.e., the number of degrees from the center to the edge (<90) [default is 60]. 541- The *scale* as 1:xxxxx or as radius/latitude where radius is distance on map in :ref:`plot-units <plt-units>` from 542 projection center to a particular oblique latitude (with **-Jf**), or map width in :ref:`plot-units <plt-units>` 543 (with **-JF**). 544 545**Description** 546 547The Gnomonic azimuthal projection is a perspective projection from the center onto a plane tangent to the surface. Its 548origin goes back to the old Greeks who used it for star maps almost 2500 years ago. The projection is neither equal-area 549nor conformal, and much distortion is introduced near the edge of the hemisphere; in fact, less than a hemisphere may be 550shown around a given center. The directions from the center of projection are true. Great circles project onto straight 551lines. Because it is mainly used for pictorial views at a small scale, only the spherical form is necessary. 552 553**Example** 554 555Using a *horizon* of 60, our example of this projection centered on 120°W/35°N can therefore be generated by the 556following :doc:`/coast` command: 557 558.. literalinclude:: /_verbatim/GMT_gnomonic.txt 559 560.. figure:: /_images/GMT_gnomonic.* 561 :width: 500 px 562 :align: center 563 564 Gnomonic azimuthal projection. 565 566 567Cylindrical projections 568----------------------- 569 570Cylindrical projections are easily recognized for their shape: maps are rectangular and meridians and parallels are 571straight lines crossing at right angles. But that is where similarities between the cylindrical projections supported 572by GMT (:ref:`Mercator <-Jm>`, :ref:`transverse Mercator <-Jt>`, :ref:`universal transverse Mercator <-Ju>`, 573:ref:`oblique Mercator <-Jo>`, :ref:`Cassini <-Jc>`, :ref:`cylindrical equidistant <-Jq>`, 574:ref:`cylindrical equal-area <-Jy>`, :ref:`Miller <-Jj>`, and :ref:`cylindrical stereographic <-Jcyl_stere>`) stops. 575Each have a different way of spacing the meridians and parallels to obtain certain desirable cartographic properties. 576 577.. _-Jm: 578 579Mercator projection (**-Jm** **-JM**) 580~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ 581 582**Syntax** 583 584 **-Jm**\|\ **M**\ [*lon0/*\ [*lat0/*]]\ *scale*\|\ *width* 585 586**Parameters** 587 588- Optionally, the central meridian (*lon0*) [default is the middle of the map]. 589- Optionally, the standard parallel for true scale (*lat0*) [default is the equator]. When supplied, the central 590 meridian (*lon0*) must be supplied as well. 591- The *scale* along the equator in :ref:`plot-units <plt-units>`/degree or as 1:xxxxx (with **-Jm**) or map *width* in 592 :ref:`plot-units <plt-units>` (with **-JM**). 593 594**Description** 595 596Probably the most famous of the various map projections, the Mercator projection takes its name from the Flemish 597cartographer Gheert Cremer, better known as Gerardus Mercator, who presented it in 1569. The projection is a cylindrical 598and conformal, with no distortion along the equator. A major navigational feature of the projection is that a line of 599constant azimuth is straight. Such a line is called a rhumb line or *loxodrome*. Thus, to sail from one point to another 600one only had to connect the points with a straight line, determine the azimuth of the line, and keep this constant 601course for the entire voyage\ [21]_. The Mercator projection has been used extensively for world maps in which the 602distortion towards the polar regions grows rather large, thus incorrectly giving the impression that, for example, 603Greenland is larger than South America. In reality, the latter is about eight times the size of Greenland. Also, the 604Former Soviet Union looks much bigger than Africa or South America. One may wonder whether this illusion has had any 605influence on U.S. foreign policy. 606 607In the regular Mercator projection, the cylinder touches the globe along the equator. Other orientations like vertical 608and oblique give rise to the :ref:`transverse Mercator <-Jt>` and :ref:`oblique Mercator <-Jo>` projections, 609respectively. We will discuss these generalizations following the regular Mercator projection. 610 611**Example** 612 613A 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: 614 615.. literalinclude:: /_verbatim/GMT_mercator.txt 616 617.. figure:: /_images/GMT_mercator.* 618 :width: 500 px 619 :align: center 620 621 Simple Mercator map. 622 623 624While this example is centered on the Dateline, one can easily choose another configuration with the **-R** option. For 625example, specify the region with **-R**-180/180/-70/70 to obtain a map centered on Greenwich. 626 627.. _-Jt: 628 629Transverse Mercator projection (**-Jt** **-JT**) 630~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ 631 632**Syntax** 633 634 **-Jt**\|\ **T**\ *lon0/*\ [*lat0/*]\ *scale*\|\ *width* 635 636**Parameters** 637 638- The central meridian (*lon0*). 639- Optionally, the latitude of origin (*lat0*) [default is the equator]. 640- The *scale* along the equator in :ref:`plot-units <plt-units>`/degree or 1:xxxxx (with **-Jt**) or map 641 *width* in :ref:`plot-units <plt-units>` (with **-JT**). 642 643You can change the map scale factor via the :term:`PROJ_SCALE_FACTOR` parameter [default is **1**]. 644 645**Description** 646 647The transverse Mercator was invented by Johann Heinrich Lambert in 1772. In this projection the cylinder touches a 648meridian along which there is no distortion. The distortion increases away from the central meridian and goes to 649infinity at 90° from center. The central meridian, each meridian 90° away from the center, and equator are straight 650lines; other parallels and meridians are complex curves. 651 652**Example** 653 654A transverse Mercator map of south-east Europe and the Middle East with 35°E as the central meridian can be obtained as 655follows: 656 657.. literalinclude:: /_verbatim/GMT_transverse_merc.txt 658 659.. figure:: /_images/GMT_transverse_merc.* 660 :width: 500 px 661 :align: center 662 663 Rectangular Transverse Mercator map. 664 665 666A global transverse Mercator map - the equivalent of the 360° Mercator map - can also be obtained as follows: 667 668.. literalinclude:: /_verbatim/GMT_TM.txt 669 670.. _GMT_TM: 671 672.. figure:: /_images/GMT_TM.* 673 :width: 450 px 674 :align: center 675 676 A global transverse Mercator map. 677 678Note that when a world map is given (indicated by **-R**\ *0/360/s/n*), the arguments are interpreted to mean oblique 679degrees, i.e., the 360° range is understood to mean the extent of the plot along the central meridian, while the "south" 680and "north" values represent how far from the central longitude we want the plot to extend. These values correspond to 681latitudes in the regular Mercator projection and must therefore be less than 90. 682 683 684 685.. _-Ju: 686 687Universal Transverse Mercator (UTM) projection (**-Ju** **-JU**) 688~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ 689 690**Syntax** 691 692 **-Ju**\|\ **U**\ *zone/*\ *scale*\|\ *width* 693 694**Parameters** 695 696- UTM *zone* (A, B, 1–60, Y, Z). Use negative values for numerical zones in the southern hemisphere or append the 697 latitude modifiers (C–H, J–N, P–X) to specify an exact UTM grid zone. 698- The *scale* along the equator in :ref:`plot-units <plt-units>`/degree or as 1:xxxxx (with **-Ju**) or map *width* in 699 :ref:`plot-units <plt-units>` (with **-JU**). 700 701**Description** 702 703A particular subset of the :ref:`transverse Mercator <-Jt>` is the Universal Transverse Mercator (UTM) which was adopted 704by the US Army for large-scale military maps. Here, the globe is divided into 60 zones between 84°S and 84°N, most of 705which are 6° wide. Each of these UTM zones have a unique central meridian. Furthermore, each zone is divided into 706latitude bands but these are not needed to specify the projection for most cases. See Figure 707:ref:`Universal Transverse Mercator <GMT_utm_zones>` for all zone designations. 708 709.. _GMT_utm_zones: 710 711.. figure:: /_images/GMT_utm_zones.* 712 :width: 700 px 713 :align: center 714 715 Universal Transverse Mercator zone layout. 716 717.. toggle:: 718 719 Here is the source script for the figure above: 720 721 .. literalinclude:: /_verbatim/GMT_utm_zones.txt 722 723In order to minimize the distortion in any given zone, a scale factor of 0.9996 has been factored into the formulae 724(although a standard, you can change this with :term:`PROJ_SCALE_FACTOR`). This makes the UTM projection a *secant* 725projection and not a *tangent* projection like the :ref:`transverse Mercator <-Jt>` above. The scale only varies by 1 726part in 1,000 from true scale at equator. The ellipsoidal projection expressions are accurate for map areas that extend 727less than 10 away from the central meridian. For larger regions we use the conformal latitude in the general spherical 728formulae instead. 729 730.. _-Jo: 731 732Oblique Mercator projection (**-Jo** **-JO**) 733~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ 734 735**Option 1 Syntax** 736 737 **-Jo**\|\ **O**\ [**a**\|\ **A**]\ *lon0/lat0/azimuth/*\ *scale*\|\ *width*\ [**+v**] 738 739**Option 1 Parameters** 740 741 - The longitude (*lon0*) and latitude (*lat0*) of projection center. 742 - The azimuth (*azimuth*) of the oblique equator. 743 - The *scale* in :ref:`plot-units <plt-units>`/degree or 1:xxxxx along oblique equator (with **-Jo**), 744 or map *width* in :ref:`plot-units <plt-units>` (with **-JO**). 745 - Optionally, append **+v** to let the oblique Equator align with the *y*-axis [default is to align with the 746 *x*-axis]. 747 748**Option 2 Syntax** 749 750 **-Jo**\|\ **O**\ [**b**\|\ **B**]\ *lon0/lat0/lon1/lat1/*\ *scale*\|\ *width*\ [**+v**] 751 752**Option 2 Parameters** 753 754 - The longitude (*lon0*) and latitude (*lat0*) of projection center. 755 - The longitude (*lon1*) and latitude (*lat1*) of a second point on oblique equator. 756 - The *scale* in :ref:`plot-units <plt-units>`/degree or 1:xxxxx along oblique equator (with **-Jo**), 757 or map *width* in :ref:`plot-units <plt-units>` (with **-JO**). 758 - Optionally, append **+v** to let the oblique Equator align with the *y*-axis [default is to align with the 759 *x*-axis]. 760 761**Option 3 Syntax** 762 763 **-Jo**\|\ **O**\ [**c**\|\ **C**]\ *lon0/lat0/lonp/latp/*\ *scale*\|\ *width*\ [**+v**] 764 765**Option 3 Parameters** 766 767 - The longitude (*lon0*) and latitude (*lat0*) of projection center. 768 - The longitude (*lonp*) and latitude (*latp*) of the projection pole. 769 - The *scale* in :ref:`plot-units <plt-units>`/degree or 1:xxxxx along oblique equator (with **-Jo**), 770 or map *width* in :ref:`plot-units <plt-units>` (with **-JO**). 771 - Optionally, append **+v** to let the oblique Equator align with the *y*-axis [default is to align with the 772 *x*-axis]. 773 774For all three definitions, the upper case **A**\|\ **B**\|\ **C** means we will allow projection poles in the southern 775hemisphere [default is to map any such poles to their antipodes in the northern hemisphere]. 776 777**Description** 778 779Oblique configurations of the cylinder give rise to the oblique Mercator projection. It is particularly useful when 780mapping regions of large lateral extent in an oblique direction. Both parallels and meridians are complex curves. The 781projection was developed in the early 1900s by several workers. 782 783**Example** 784 785An oblique view of some Caribbean islands using Option 3 can be obtained as follows: 786 787.. literalinclude:: /_verbatim/GMT_obl_merc.txt 788 789.. figure:: /_images/GMT_obl_merc.* 790 :width: 500 px 791 :align: center 792 793 Oblique Mercator map using **-JOc**. We make it clear which direction is North by adding a star rose with the **-Td** 794 option. 795 796 797Note that we define our region using the rectangular system described earlier. If we do not append **+r** to the **-R** 798string then the information provided with the **-R** option is assumed to be oblique degrees about the projection center 799rather than the usual geographic coordinates. This interpretation is chosen since in general the parallels and meridians 800are not very suitable as map boundaries. 801 802When working with oblique projections such as here, it is often much more convenient to specify the map domain in the 803projected coordinates relative to the map center. The figure below shows two views of New Zealand using the oblique 804Mercator projection that in both cases specifies the region using **-R**\ -1000/1000/-500/500\ **+uk**. The unit **k** 805means the following bounds are in projected km and we let GMT determine the geographic coordinates of the two diagonal 806corners internally. 807 808.. figure:: /_images/GMT_obl_nz.* 809 :width: 600 px 810 :align: center 811 812 (left) Oblique view of New Zealand centered on its geographical center (Nelson) indicated by the white circle for an 813 oblique Equator with azimuth 35. This resulted in the argument **-JOa**\ 173:17:02E/41:16:15S/35/3i. The map is 814 2000 km by 1000 km and the Cartesian coordinate system in the projected units are indicated by the bold axes. The 815 blue circle is the point (40S,180E) and it has projected coordinates (*x* = 426.2, *y* = -399.7). 816 (right) Same dimensions but now specifying an azimuth of 215, yielding a projection pole in the southern hemisphere 817 (hence we used **-JOA** to override the restriction, i.e., **-JOA**\ 173:17:02E/41:16:15S/215/3i.) 818 The projected coordinate system is still aligned as before but the Earth has been rotated 180 degrees. The blue 819 point now has projected coordinates (*x* = -426.2, *y* = 399.7). 820 821.. toggle:: 822 823 Here is the source script for the figure above: 824 825 .. literalinclude:: /_verbatim/GMT_obl_nz.txt 826 827The oblique Mercator projection will by default arrange the output so that the oblique Equator becomes the new 828horizontal, positive *x*-axis. For features with an orientation more north-south than east-west, it may be preferable 829to align the oblique Equator with the vertical, positive *y*-axis instead. This configuration is selected by appending 830**+v** to the **-J** projection option. The example below shows this behaviour. 831 832.. figure:: /_images/GMT_obl_baja.* 833 :width: 300 px 834 :align: center 835 836 Oblique view of Baja California using the vertical oblique Equator modifier. This plot 837 resulted from the argument **-JOa**\ 120W/25N/-30/6c\ **+v**\ . 838 839.. toggle:: 840 841 Here is the source script for the figure above: 842 843 .. literalinclude:: /_verbatim/GMT_obl_baja.txt 844 845.. _-Jc: 846 847Cassini cylindrical projection (**-Jc** **-JC**) 848~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ 849 850**Syntax** 851 852 **-Jc**\|\ **C**\ *lon0/lat0/scale*\|\ *width* 853 854**Parameters** 855 856 - The longitude (*lon0*) and latitude (*lat0*) of the central point. 857 - The *scale* in :ref:`plot-units <plt-units>`/degree or as 1:xxxxx (with **-Jc**) or map *width* in 858 :ref:`plot-units <plt-units>` (with **-JC**). 859 860**Description** 861 862This cylindrical projection was developed in 1745 by César-François Cassini de Thury for the survey of France. It is 863occasionally called Cassini-Soldner since the latter provided the more accurate mathematical analysis that led to the 864development of the ellipsoidal formulae. The projection is neither conformal nor equal-area, and behaves as a compromise 865between the two end-members. The distortion is zero along the central meridian. It is best suited for mapping regions of 866north-south extent. The central meridian, each meridian 90° away, and equator are straight lines; all other meridians 867and parallels are complex curves. 868 869**Example** 870 871A detailed map of the island of Sardinia centered on the 8°45'E meridian using the Cassini projection can be obtained by 872as follows: 873 874.. literalinclude:: /_verbatim/GMT_cassini.txt 875 876.. figure:: /_images/GMT_cassini.* 877 :width: 400 px 878 :align: center 879 880 Cassini map over Sardinia. 881 882 883As with the previous projections, the user can choose between a rectangular boundary (used here) or a geographical 884(WESN) boundary. 885 886.. _-Jq: 887 888Cylindrical equidistant projection (**-Jq** **-JQ**) 889~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ 890 891**Syntax** 892 893 **-Jq**\|\ **Q**\ [*lon0/*\ [*lat0/*]]\ *scale*\|\ *width* 894 895**Parameters** 896 897- Optionally, the central meridian (*lon0*) [default is the middle of the map map]. 898- Optionally, the standard parallel (*lat0*) [default is the equator]. When supplied, the central meridian (*lon0*) 899 must be supplied as well. 900- The *scale* in :ref:`plot-units <plt-units>`/degree or as 1:xxxxx (with **-Jq**) or map *width* in 901 :ref:`plot-units <plt-units>` (with **-JQ**). 902 903**Description** 904 905This simple cylindrical projection is really a linear scaling of longitudes and latitudes. The most common form is the 906Plate Carrée projection, where the scaling of longitudes and latitudes is the same. All meridians and parallels are 907straight lines. 908 909Different relative scalings of longitudes and latitudes can be obtained by selecting a standard parallel different from 910the equator. Some selections for standard parallels have practical properties as shown in Table :ref:`JQ <tbl-JQ>`. 911 912.. _tbl-JQ: 913 914+-----------------------------------------------------+--------+ 915+=====================================================+========+ 916| Grafarend and Niermann, minimum linear distortion | 61.7° | 917+-----------------------------------------------------+--------+ 918| Ronald Miller Equirectangular | 50.5° | 919+-----------------------------------------------------+--------+ 920| Ronald Miller, minimum continental distortion | 43.5° | 921+-----------------------------------------------------+--------+ 922| Grafarend and Niermann | 42° | 923+-----------------------------------------------------+--------+ 924| Ronald Miller, minimum overall distortion | 37.5° | 925+-----------------------------------------------------+--------+ 926| Plate Carrée, Simple Cylindrical, Plain/Plane | 0° | 927+-----------------------------------------------------+--------+ 928 929**Example** 930 931A world map centered on the dateline using this projection can be obtained as follows: 932 933.. literalinclude:: /_verbatim/GMT_equi_cyl.txt 934 935.. figure:: /_images/GMT_equi_cyl.* 936 :width: 500 px 937 :align: center 938 939 World map using the Plate Carrée projection. 940 941.. _-Jy: 942 943Cylindrical equal-area projections (**-Jy** **-JY**) 944~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ 945 946**Syntax** 947 948 **-Jy**\|\ **Y**\ [*lon0/*\ [*lat0/*]]\ *scale*\|\ *width* 949 950**Parameters** 951 952- Optionally, the central meridian (*lon0*) [default is the middle of the map]. 953- Optionally, the standard parallel (*lat0*) [default is the equator]. When supplied, the central meridian (*lon0*) 954 must be supplied as well. 955- The *scale* in :ref:`plot-units <plt-units>`/degree or as 1:xxxxx (with **-Jy**) or map *width* in 956 :ref:`plot-units <plt-units>` (with **-JY**) 957 958**Description** 959 960This cylindrical projection is actually several projections, depending on what latitude is selected as the standard 961parallel. However, they are all equal area and hence non-conformal. All meridians and parallels are straight lines. 962 963While you may choose any value for the standard parallel and obtain your own personal projection, there are seven 964choices of standard parallels that result in known (or named) projections. These are listed in Table :ref:`JY <tbl-JY>`. 965 966.. _tbl-JY: 967 968+-------------------+---------------------+ 969+===================+=====================+ 970| Balthasart | 50° | 971+-------------------+---------------------+ 972| Gall | 45° | 973+-------------------+---------------------+ 974| Hobo-Dyer | 37°30' (= 37.5°) | 975+-------------------+---------------------+ 976| Trystan Edwards | 37°24' (= 37.4°) | 977+-------------------+---------------------+ 978| Caster | 37°04' (= 37.0666°) | 979+-------------------+---------------------+ 980| Behrman | 30° | 981+-------------------+---------------------+ 982| Lambert | 0° | 983+-------------------+---------------------+ 984 985**Example** 986 987A world map centered on the 35°E meridian using the Behrman projection (Figure 988:ref:`Behrman cylindrical projection <GMT_general_cyl>`) can be obtained as follows: 989 990.. literalinclude:: /_verbatim/GMT_general_cyl.txt 991 992.. _GMT_general_cyl: 993 994.. figure:: /_images/GMT_general_cyl.* 995 :width: 600 px 996 :align: center 997 998 World map using the Behrman cylindrical equal-area projection. 999 1000 1001As one can see there is considerable distortion at high latitudes since the poles map into lines. 1002 1003.. _-Jj: 1004 1005Miller Cylindrical projection (**-Jj** **-JJ**) 1006~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ 1007 1008**Syntax** 1009 1010 **-Jj**\|\ **J**\ [*lon0/*]\ *scale*\|\ *width* 1011 1012**Parameters** 1013 1014- Optionally, the central meridian (*lon0*) [default is the middle of the map]. 1015 1016- The *scale* in :ref:`plot-units <plt-units>`/degree or as 1:xxxxx (with **-Jj**) or map *width* in 1017 :ref:`plot-units <plt-units>` (with **-JJ**). 1018 1019**Description** 1020 1021This cylindrical projection, presented by Osborn Maitland Miller of the American Geographic Society in 1942, is neither 1022equal nor conformal. All meridians and parallels are straight lines. The projection was designed to be a compromise 1023between Mercator and other cylindrical projections. Specifically, Miller spaced the parallels by using Mercator's 1024formula with 0.8 times the actual latitude, thus avoiding the singular poles; the result was then divided by 0.8. There 1025is only a spherical form for this projection. 1026 1027**Example** 1028 1029A world map centered on the 90°E meridian at a map scale of 1:400,000,000 (Figure :ref:`Miller projection <GMT_miller>`) 1030can be obtained as follows: 1031 1032.. literalinclude:: /_verbatim/GMT_miller.txt 1033 1034.. _GMT_miller: 1035 1036.. figure:: /_images/GMT_miller.* 1037 :width: 500 px 1038 :align: center 1039 1040 World map using the Miller cylindrical projection. 1041 1042.. _-Jcyl_stere: 1043 1044Cylindrical stereographic projections (**-Jcyl_stere** **-JCyl_stere**) 1045~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ 1046 1047**Syntax** 1048 1049 **-Jcyl_stere**\|\ **Cyl_stere**\ /[*lon0/*\ [*lat0/*]]\ *scale*\|\ *width* 1050 1051**Parameters** 1052 1053- Optionally, the central meridian (*lon0*) [default is the middle of the map]. 1054- Optionally, the standard parallel (*lat0*) [default is the Equator]. When used, central meridian (*lon0*) needs to be 1055 given as well. 1056- The *scale* in :ref:`plot-units <plt-units>`/degree or as 1:xxxxx (with **-Jcyl_stere**) or map *width* in 1057 :ref:`plot-units <plt-units>` (with **-JCyl_stere**). 1058 1059**Description** 1060 1061The cylindrical stereographic projections are certainly not as notable as other cylindrical projections, but are still 1062used because of their relative simplicity and their ability to overcome some of the downsides of other cylindrical 1063projections, like extreme distortions of the higher latitudes. The stereographic projections are perspective 1064projections, projecting the sphere onto a cylinder in the direction of the antipodal point on the equator. The cylinder 1065crosses the sphere at two standard parallels, equidistant from the equator. 1066 1067Some of the selections of the standard parallel are named for the cartographer or publication that popularized the 1068projection (Table :ref:`JCylstere <tbl-JCylstere>`). 1069 1070.. _tbl-JCylstere: 1071 1072+---------------------------------------------------------+-------------+ 1073+=========================================================+=============+ 1074| Miller's modified Gall | 66.159467° | 1075+---------------------------------------------------------+-------------+ 1076| Kamenetskiy's First | 55° | 1077+---------------------------------------------------------+-------------+ 1078| Gall's stereographic | 45° | 1079+---------------------------------------------------------+-------------+ 1080| Bolshoi Sovietskii Atlas Mira or Kamenetskiy's Second | 30° | 1081+---------------------------------------------------------+-------------+ 1082| Braun's cylindrical | 0° | 1083+---------------------------------------------------------+-------------+ 1084 1085**Example** 1086 1087A map of the world, centered on the Greenwich meridian, using the Gall's stereographic projection (standard parallel is 108845°, Figure :ref:`Gall's stereographic projection <GMT_gall_stereo>`), can be obtained as follows: 1089 1090.. literalinclude:: /_verbatim/GMT_gall_stereo.txt 1091 1092.. _GMT_gall_stereo: 1093 1094.. figure:: /_images/GMT_gall_stereo.* 1095 :width: 500 px 1096 :align: center 1097 1098 World map using Gall's stereographic projection. 1099 1100 1101Miscellaneous projections 1102------------------------- 1103 1104GMT supports eight common projections for global presentation of data or models. These are the :ref:`Hammer <-Jh>`, 1105:ref:`Mollweide <-Jw>`, :ref:`Winkel Tripel <-Jr>`, :ref:`Robinson <-Jn>`, :ref:`Eckert IV and VI <-Jk>`, 1106:ref:`Sinusoidal <-Ji>`, and :ref:`Van der Grinten <-Jv>` projections. Due to the small scale used for global maps these 1107projections all use the spherical approximation rather than more elaborate elliptical formulae. 1108 1109In all cases, the specification of the central meridian can be skipped. The default is the middle of the longitude 1110range of the plot, specified by the (**-R**) option. 1111 1112.. _-Jh: 1113 1114Hammer projection (**-Jh** **-JH**) 1115~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ 1116 1117**Syntax** 1118 1119 **-Jh**\|\ **H**\ [*lon0/*]\ *scale*\|\ *width* 1120 1121**Parameters** 1122 1123- The central meridian (*lon0*) [default is the middle of the map]. 1124- The *scale* along equator in :ref:`plot-units <plt-units>`/degree or as 1:xxxxx (with **-Jh**) or map *width* in 1125 :ref:`plot-units <plt-units>` (with **-JH**). 1126 1127**Description** 1128 1129The equal-area Hammer projection, first presented by the German mathematician Ernst von Hammer in 1892, is also known as 1130Hammer-Aitoff (the Aitoff projection looks similar, but is not equal-area). The border is an ellipse, equator and 1131central meridian are straight lines, while other parallels and meridians are complex curves. 1132 1133**Example** 1134 1135A view of the Pacific ocean using the Dateline as central meridian can be generated thus: 1136 1137.. literalinclude:: /_verbatim/GMT_hammer.txt 1138 1139.. figure:: /_images/GMT_hammer.* 1140 :width: 500 px 1141 :align: center 1142 1143 World map using the Hammer projection. 1144 1145.. _-Jw: 1146 1147Mollweide projection (**-Jw** **-JW**) 1148~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ 1149 1150**Syntax** 1151 1152 **-Jw**\|\ **W**\ [*lon0/*]\ *scale*\|\ *width* 1153 1154**Parameters** 1155 1156- The central meridian (*lon0*) [default is the middle of the map]. 1157- The *scale* along equator in :ref:`plot-units <plt-units>`/degree or as 1:xxxxx (with **-Jw**) or map *width* in 1158 :ref:`plot-units <plt-units>` (with **-JW**). 1159 1160**Description** 1161 1162This pseudo-cylindrical, equal-area projection was developed by the German mathematician and astronomer Karl Brandan 1163Mollweide in 1805. Parallels are unequally spaced straight lines with the meridians being equally spaced elliptical 1164arcs. The scale is only true along latitudes 40°44' north and south. The projection is used mainly for global maps 1165showing data distributions. It is occasionally referenced under the name *homalographic* projection. 1166 1167**Example** 1168 1169An example centered on Greenwich can be generated thus: 1170 1171.. literalinclude:: /_verbatim/GMT_mollweide.txt 1172 1173.. figure:: /_images/GMT_mollweide.* 1174 :width: 500 px 1175 :align: center 1176 1177 World map using the Mollweide projection. 1178 1179.. _-Jr: 1180 1181Winkel Tripel projection (**-Jr** **-JR**) 1182~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ 1183 1184**Syntax** 1185 1186 **-Jr**\|\ **R**\ [*lon0/*]\ *scale*\|\ *width* 1187 1188**Parameters** 1189 1190- The central meridian (*lon0*) [default is the middle of the map]. 1191- The *scale* along equator in :ref:`plot-units <plt-units>`/degree or as 1:xxxxx (with **-Jr**) or map *width* in 1192 :ref:`plot-units <plt-units>` (with **-JR**). 1193 1194**Description** 1195 1196In 1921, the German mathematician Oswald Winkel created a projection that was to strike a compromise between the 1197properties of three elements (area, angle and distance). The German word "tripel" refers to this junction of where each 1198of these elements are least distorted when plotting global maps. The projection was popularized when Bartholomew and Son 1199started to use it in its world-renowned "The Times Atlas of the World" in the mid-20th century. In 1998, the National 1200Geographic Society made the Winkel Tripel as its map projection of choice for global maps. 1201 1202Naturally, this projection is neither conformal, nor equal-area. Central meridian and equator are straight lines; other 1203parallels and meridians are curved. The projection is obtained by averaging the coordinates of the Equidistant 1204Cylindrical and Aitoff (not Hammer-Aitoff) projections. The poles map into straight lines 0.4 times the length of 1205equator. 1206 1207**Example** 1208 1209Centered on Greenwich, the example in Figure :ref:`Winkel Tripel projection <GMT_winkel>` was created by this command: 1210 1211.. literalinclude:: /_verbatim/GMT_winkel.txt 1212 1213.. _GMT_winkel: 1214 1215.. figure:: /_images/GMT_winkel.* 1216 :width: 500 px 1217 :align: center 1218 1219 World map using the Winkel Tripel projection. 1220 1221.. _-Jn: 1222 1223Robinson projection (**-Jn** **-JN**) 1224~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ 1225 1226**Syntax** 1227 1228 **-Jn**\|\ **N**\ [*lon0/*]\ *scale*\|\ *width* 1229 1230**Parameters** 1231 1232- The central meridian (*lon0*) [default is the middle of the map]. 1233- The *scale* along equator in :ref:`plot-units <plt-units>`/degree or as 1:xxxxx (with **-Jn**) or map *width* in 1234 :ref:`plot-units <plt-units>` (with **-JN**). 1235 1236**Description** 1237 1238The Robinson projection, presented by the American geographer and cartographer Arthur H. Robinson in 1963, is a modified 1239cylindrical projection that is neither conformal nor equal-area. Central meridian and all parallels are straight lines; 1240other meridians are curved. It uses lookup tables rather than analytic expressions to make the world map "look" 1241right\ [22]_. The scale is true along latitudes 38. The projection was originally developed for use by Rand McNally and 1242is currently used by the National Geographic Society. 1243 1244**Example** 1245 1246Again centered on Greenwich, the example below was created by this command: 1247 1248.. literalinclude:: /_verbatim/GMT_robinson.txt 1249 1250.. figure:: /_images/GMT_robinson.* 1251 :width: 500 px 1252 :align: center 1253 1254 World map using the Robinson projection. 1255 1256.. _-Jk: 1257 1258Eckert IV and VI projection (**-Jk** **-JK**) 1259~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ 1260 1261**Syntax** 1262 1263 **-Jk**\|\ **K**\ **f**\ [*lon0/*]\ *scale*\|\ *width* (Eckert IV) 1264 **-Jk**\|\ **K**\ [**s**][*lon0/*]\ *scale*\|\ *width* (Eckert VI) 1265 1266**Parameters** 1267 1268- The central meridian (*lon0*) [default is the middle of the map]. 1269- The *scale* along equator in :ref:`plot-units <plt-units>`/degree or as 1:xxxxx (with **-Jk**) or map *width* in 1270 :ref:`plot-units <plt-units>` (with **-JK**). 1271 1272**Description** 1273 1274The Eckert IV and VI projections, presented by the German cartographer Max Eckert-Greiffendorff in 1906, are 1275pseudo-cylindrical equal-area projections. Central meridian and all parallels are straight lines; other meridians are 1276equally spaced elliptical arcs (IV) or sinusoids (VI). The scale is true along latitudes 40°30' (IV) and 49°16' (VI). 1277Their main use is in thematic world maps. To select Eckert IV you must use **-JKf** (**f** for "four") while Eckert VI 1278is selected with **-JKs** (**s** for "six"). If no modifier is given it defaults to Eckert VI. 1279 1280**Examples** 1281 1282Centered on the Dateline, the Eckert IV example below was created by this command: 1283 1284.. literalinclude:: /_verbatim/GMT_eckert4.txt 1285 1286.. figure:: /_images/GMT_eckert4.* 1287 :width: 500 px 1288 :align: center 1289 1290 World map using the Eckert IV projection. 1291 1292 1293The same script, with **s** instead of **f**, yields the Eckert VI map: 1294 1295.. figure:: /_images/GMT_eckert6.* 1296 :width: 500 px 1297 :align: center 1298 1299 World map using the Eckert VI projection. 1300 1301.. toggle:: 1302 1303 Here is the source script for the figure above: 1304 1305 .. literalinclude:: /_verbatim/GMT_eckert6.txt 1306 1307.. _-Ji: 1308 1309Sinusoidal projection (**-Ji** **-JI**) 1310~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ 1311 1312**Syntax** 1313 1314 **-Ji**\|\ **I**\ [*lon0/*]\ *scale*\|\ *width* 1315 1316**Parameters** 1317 1318- The central meridian (*lon0*) [default is the middle of the map]. 1319- The *scale* along equator in :ref:`plot-units <plt-units>`/degree or as 1:xxxxx (with **-Ji**) or map *width* in 1320 :ref:`plot-units <plt-units>` (with **-JI**). 1321 1322**Description** 1323 1324The sinusoidal projection is one of the oldest known projections, is equal-area, and has been used since the mid-16th 1325century. It has also been called the "Equal-area Mercator" projection. The central meridian is a straight line; all 1326other meridians are sinusoidal curves. Parallels are all equally spaced straight lines, with scale being true along all 1327parallels (and central meridian). 1328 1329**Examples** 1330 1331A simple world map using the sinusoidal projection is therefore obtained by 1332 1333.. literalinclude:: /_verbatim/GMT_sinusoidal.txt 1334 1335.. figure:: /_images/GMT_sinusoidal.* 1336 :width: 500 px 1337 :align: center 1338 1339 World map using the Sinusoidal projection. 1340 1341 1342To reduce distortion of shape the interrupted sinusoidal projection was introduced in 1927. Here, three symmetrical 1343segments are used to cover the entire world. Traditionally, the interruptions are at 160°W, 20°W, and 60°E. To make the 1344interrupted map we must call :doc:`/coast` for each segment and superpose the results. To produce an interrupted world 1345map (with the traditional boundaries just mentioned) that is 14.4 cm wide we use the scale 14.4/360 = 0.04 and offset 1346the subsequent plots horizontally by their widths (140\ :math:`\cdot`\ 0.04 and 80\ :math:`\cdot`\ 0.04): 1347 1348.. literalinclude:: /_verbatim/GMT_sinus_int.txt 1349 1350.. figure:: /_images/GMT_sinus_int.* 1351 :width: 500 px 1352 :align: center 1353 1354 World map using the Interrupted Sinusoidal projection. 1355 1356 1357The usefulness of the interrupted sinusoidal projection is basically limited to display of global, discontinuous data 1358distributions like hydrocarbon and mineral resources, etc. 1359 1360.. _-Jv: 1361 1362Van der Grinten projection (**-Jv** **-JV**) 1363~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ 1364 1365**Syntax** 1366 1367 **-Jv**\|\ **V**\ [*lon0/*]\ *scale*\|\ *width* 1368 1369**Parameters** 1370 1371- The central meridian (*lon0*) [default is the middle of the map]. 1372- The *scale* along equator in :ref:`plot-units <plt-units>`/degree or as 1:xxxxx (with **-Jv**) or map *width* in 1373 :ref:`plot-units <plt-units>` (with **-JV**). 1374 1375**Description** 1376 1377The Van der Grinten projection, presented by Alphons J. van der Grinten in 1904, is neither equal-area nor conformal. 1378Central meridian and Equator are straight lines; other meridians are arcs of circles. The scale is true along the 1379Equator only. Its main use is to show the entire world enclosed in a circle. 1380 1381**Example** 1382 1383Centered on the Dateline, the example below was created by this command: 1384 1385.. literalinclude:: /_verbatim/GMT_grinten.txt 1386 1387.. figure:: /_images/GMT_grinten.* 1388 :width: 400 px 1389 :align: center 1390 1391 World map using the Van der Grinten projection. 1392 1393Footnotes 1394--------- 1395 1396.. [20] 1397 Snyder, J. P., 1987, Map Projections A Working Manual, U.S. 1398 Geological Survey Prof. Paper 1395. 1399 1400.. [21] 1401 This is, however, not the shortest distance. It is given by the great 1402 circle connecting the two points. 1403 1404.. [22] 1405 Robinson provided a table of *y*-coordinates for latitudes 1406 every 5. To project values for intermediate latitudes one must 1407 interpolate the table. Different interpolants may result in slightly 1408 different maps. GMT uses the 1409 interpolant selected by the parameter :term:`GMT_INTERPOLANT` in the 1410 file. 1411