1Cartography(3) User Contributed Perl Documentation Cartography(3)
2
6 PDL::Transform::Cartography - Useful cartographic projections
7
9 # make a Mercator map of Earth
10 use PDL::Transform::Cartography;
11 $x = earth_coast();
12 $x = graticule(10,2)->glue(1,$x);
13 $t = t_mercator;
14 $w = pgwin(xs);
15 $w->lines($t->apply($x)->clean_lines());
16
18 PDL::Transform::Cartography includes a variety of useful cartographic
19 and observing projections (mappings of the surface of a sphere),
20 including reprojected observer coordinates. See PDL::Transform for
21 more information about image transforms in general.
22 Cartographic transformations are used for projecting not just
24 terrestrial maps, but also any nearly spherical surface including the
25 Sun, the Celestial sphere, various moons and planets, distant stars,
26 etc. They also are useful for interpreting scientific images, which
27 are themselves generally projections of a sphere onto a flat focal
28 plane (e.g. the t_gnomonic projection).
29 Unless otherwise noted, all the transformations in this file convert
31 from (theta,phi) coordinates on the unit sphere (e.g. (lon,lat) on a
32 planet or (RA,dec) on the celestial sphere) into some sort of projected
33 coordinates, and have inverse transformations that convert back to
34 (theta,phi). This is equivalent to working from the equidistant
35 cylindrical (or "plate caree") projection, if you are a cartography
36 wonk.
37 The projected coordinates are generally in units of body radii
39 (radians), so that multiplying the output by the scale of the map
40 yields physical units that are correct wherever the scale is correct
41 for that projection. For example, areas should be correct everywhere
42 in the authalic projections; and linear scales are correct along
43 meridians in the equidistant projections and along the standard
44 parallels in all the projections.
45 The transformations that are authalic (equal-area), conformal (equal-
47 angle), azimuthal (circularly symmetric), or perspective (true
48 perspective on a focal plane from some viewpoint) are marked. The
49 first two categories are mutually exclusive for all but the "unit
50 sphere" 3-D projection.
51 Extra dimensions tacked on to each point to be transformed are, in
53 general, ignored. That is so that you can add on an extra index to
54 keep track of pen color. For example, earth_coast returns a 3x<n>
55 piddle containing (lon, lat, pen) at each list location. Transforming
56 the vector list retains the pen value as the first index after the
57 dimensional directions.
58
60 Unless otherwise noted, the transformations and miscellaneous
61 information in this section are taken from Snyder & Voxland 1989: "An
62 Album of Map Projections", US Geological Survey Professional Paper
63 1453, US Printing Office (Denver); and from Snyder 1987: "Map
64 Projections - A Working Manual", US Geological Survey Professional
65 Paper 1395, US Printing Office (Denver, USA). You can obtain your own
66 copy of both by contacting the U.S. Geological Survey, Federal Center,
67 Box 25425, Denver, CO 80225 USA.
68 The mathematics of cartography have a long history, and the details are
70 far trickier than the broad overview. For terrestrial (and, in
71 general, planetary) cartography, the best reference datum is not a
72 sphere but an oblate ellipsoid due to centrifugal force from the
73 planet's rotation. Furthermore, because all rocky planets, including
74 Earth, have randomly placed mass concentrations that affect the
75 gravitational field, the reference gravitational isosurface (sea level
76 on Earth) is even more complex than an ellipsoid and, in general,
77 different ellipsoids have been used for different locations at the same
78 time and for the same location at different times.
79 The transformations in this package use a spherical datum and hence
81 include global distortion at about the 0.5% level for terrestrial maps
82 (Earth's oblateness is ~1/300). This is roughly equal to the
83 dimensional precision of physical maps printed on paper (due to
84 stretching and warping of the paper) but is significant at larger
85 scales (e.g. for regional maps). If you need more precision than that,
86 you will want to implement and use the ellipsoidal transformations from
87 Snyder 1987 or another reference work on geodesy. A good name for that
88 package would be "...::Cartography::Geodetic".
89
91 Cartographic transformations are useful for interpretation of
92 scientific images, as all cameras produce projections of the celestial
93 sphere onto the focal plane of the camera. A simple (single-element)
94 optical system with a planar focal plane generates gnomonic images --
95 that is to say, gnomonic projections of a portion of the celestial
96 sphere near the paraxial direction. This is the projection that most
97 consumer grade cameras produce.
98 Magnification in an optical system changes the angle of incidence of
100 the rays on the focal plane for a given angle of incidence at the
101 aperture. For example, a 10x telescope with a 2 degree field of view
102 exhibits the same gnomonic distortion as a simple optical system with a
103 20 degree field of view. Wide-angle optics typically have
104 magnification less than 1 ('fisheye lenses'), reducing the gnomonic
105 distortion considerably but introducing "equidistant azimuthal"
106 distortion -- there's no such thing as a free lunch!
107 Because many solar-system objects are spherical,
109 PDL::Transform::Cartography includes perspective projections for
110 producing maps of spherical bodies from perspective views. Those
111 projections are "t_vertical" and "t_perspective". They map between
112 (lat,lon) on the spherical body and planar projected coordinates at the
113 viewpoint. "t_vertical" is the vertical perspective projection given
114 by Snyder, but "t_perspective" is a fully general perspective
115 projection that also handles magnification correction.
116
118 Oblique projections rotate the sphere (and graticule) to an arbitrary
119 angle before generating the projection; transverse projections rotate
120 the sphere exactly 90 degrees before generating the projection.
121 Most of the projections accept the following standard options, useful
123 for making transverse and oblique projection maps.
124 o, origin, Origin [default (0,0,0)]
126 The origin of the oblique map coordinate system, in (old-theta, old-
127 phi) coordinates.
128 r, roll, Roll [default 0.0]
130 The roll angle of the sphere about the origin, measured CW from (N =
131 up) for reasonable values of phi and CW from (S = up) for
132 unreasonable values of phi. This is equivalent to observer roll
133 angle CCW from the same direction.
134 u, unit, Unit [default 'degree']
136 This is the name of the angular unit to use in the lon/lat
137 coordinate system.
138 b, B
140 The "B" angle of the body -- used for extraterrestrial maps.
141 Setting this parameter is exactly equivalent to setting the phi
142 component of the origin, and in fact overrides it.
143 l,L
145 The longitude of the central meridian as observed -- used for
146 extraterrestrial maps. Setting this parameter is exactly equivalent
147 to setting the theta component of the origin, and in fact overrides
148 it.
149 p,P
151 The "P" (or position) angle of the body -- used for extraterrestrial
152 maps. This parameter is a synonym for the roll angle, above.
153 bad, Bad, missing, Missing [default nan]
155 This is the value that missing points get. Mainly useful for the
156 inverse transforms. (This should work fine if set to BAD, if you
157 have bad-value support compiled in). The default nan is asin(1.2),
158 calculated at load time.
159
161 Draw a Mercator map of the world on-screen:
162 $w = pgwin(xs);
164 $w->lines(earth_coast->apply(t_mercator)->clean_lines);
165 Here, "earth_coast()" returns a 3xn piddle containing (lon, lat, pen)
167 values for the included world coastal outline; "t_mercator" converts
168 the values to projected Mercator coordinates, and "clean_lines" breaks
169 lines that cross the 180th meridian.
170 Draw a Mercator map of the world, with lon/lat at 10 degree intervals:
172 $w = pgwin(xs)
174 $x = earth_coast()->glue(1,graticule(10,1));
175 $w->lines($x->apply(t_mercator)->clean_lines);
176 This works just the same as the first example, except that a map
178 graticule has been applied with interline spacing of 10 degrees lon/lat
179 and inter-vertex spacing of 1 degree (so that each meridian contains
180 181 points, and each parallel contains 361 points).
181
183 Currently angular conversions are rather simpleminded. A list of
184 common conversions is present in the main constructor, which inserts a
185 conversion constant to radians into the {params} field of the new
186 transform. Something like Math::Convert::Units should be used instead
187 to generate the conversion constant.
188 A cleaner higher-level interface is probably needed (see the examples);
190 for example, earth_coast could return a graticule if asked, instead of
191 needing one to be glued on.
192 The class structure is somewhat messy because of the varying needs of
194 the different transformations. PDL::Transform::Cartography is a base
195 class that interprets the origin options and sets up the basic
196 machinery of the Transform. The conic projections have their own
197 subclass, PDL::Transform::Conic, that interprets the standard
198 parallels. Since the cylindrical and azimuthal projections are pretty
199 simple, they are not subclassed.
200 The perl 5.6.1 compiler is quite slow at adding new classes to the
202 structure, so it does not makes sense to subclass new transformations
203 merely for the sake of pedantry.
204
206 Copyright 2002, Craig DeForest (deforest@boulder.swri.edu). This
207 module may be modified and distributed under the same terms as PDL
208 itself. The module comes with NO WARRANTY.
209 The included digital world map is derived from the 1987 CIA World Map,
211 translated to ASCII in 1988 by Joe Dellinger (geojoe@freeusp.org) and
212 simplified in 1995 by Kirk Johnson (tuna@indra.com) for the program
213 XEarth. The map comes with NO WARRANTY. An ASCII version of the map,
214 and a sample PDL function to read it, may be found in the Demos
215 subdirectory of the PDL source distribution.
216
218 The module exports both transform constructors ('t_<foo>') and some
219 auxiliary functions (no leading 't_').
220 graticule
222 $lonlatp = graticule(<grid-spacing>,<line-segment-size>);
223 $lonlatp = graticule(<grid-spacing>,<line-segment-size>,1);
224 (Cartography) PDL constructor - generate a lat/lon grid.
226 Returns a grid of meridians and parallels as a list of vectors suitable
228 for sending to PDL::Graphics::PGPLOT::Window::lines for plotting. The
229 grid is in degrees in (theta, phi) coordinates -- this is (E lon, N
230 lat) for terrestrial grids or (RA, dec) for celestial ones. You must
231 then transform the graticule in the same way that you transform the
232 map.
233 You can attach the graticule to a vector map using the syntax:
235 $out = graticule(10,2)->glue(1,$map);
237 In array context you get back a 2-element list containing a piddle of
239 the (theta,phi) pairs and a piddle of the pen values (1 or 0) suitable
240 for calling PDL::Graphics::PGPLOT::Window::lines. In scalar context
241 the two elements are combined into a single piddle.
242 The pen values associated with the graticule are negative, which will
244 cause PDL::Graphics::PGPLOT::Window::lines to plot them as hairlines.
245 If a third argument is given, it is a hash of options, which can be:
247 nan - if true, use two columns instead of three, and separate lines
249 with a 'nan' break
250 lonpos - if true, all reported longitudes are positive (0 to 360)
251 instead of (-180 to 180).
252 dup - if true, the meridian at the far boundary is duplicated.
253 earth_coast
255 $x = earth_coast()
256 (Cartography) PDL constructor - coastline map of Earth
258 Returns a vector coastline map based on the 1987 CIA World Coastline
260 database (see author information). The vector coastline data are in
261 plate caree format so they can be converted to other projections via
262 the apply method and cartographic transforms, and are suitable for
263 plotting with the lines method in the PGPLOT output library: the first
264 dimension is (X,Y,pen) with breaks having a pen value of 0 and
265 hairlines having negative pen values. The second dimension threads
266 over all the points in the data set.
267 The vector map includes lines that pass through the antipodean
269 meridian, so if you want to plot it without reprojecting, you should
270 run it through clean_lines first:
271 $w = pgwin();
273 $w->lines(earth_coast->clean_lines); # plot plate caree map of world
274 $w->lines(earth_coast->apply(t_gnomonic))# plot gnomonic map of world
275 "earth_coast" is just a quick-and-dirty way of loading the file
277 "earth_coast.vec.fits" that is part of the normal installation tree.
278 earth_image
280 $rgb = earth_image()
281 (Cartography) PDL constructor - RGB pixel map of Earth
283 Returns an RGB image of Earth based on data from the MODIS instrument
285 on the NASA EOS/Terra satellite. (You can get a full-resolution image
286 from <http://earthobservatory.nasa.gov/Newsroom/BlueMarble/>). The
287 image is a plate caree map, so you can convert it to other projections
288 via the map method and cartographic transforms.
289 This is just a quick-and-dirty way of loading the earth-image files
291 that are distributed along with PDL.
292 clean_lines
294 $x = clean_lines(t_mercator->apply(scalar(earth_coast())));
295 $x = clean_lines($line_pen, [threshold]);
296 $x = $lines->clean_lines;
297 (Cartography) PDL method - remove projection irregularities
299 "clean_lines" massages vector data to remove jumps due to singularities
301 in the transform.
302 In the first (scalar) form, $line_pen contains both (X,Y) points and
304 pen values suitable to be fed to lines: in the second (list) form,
305 $lines contains the (X,Y) points and $pen contains the pen values.
306 "clean_lines" assumes that all the outline polylines are local -- that
308 is to say, there are no large jumps. Any jumps larger than a threshold
309 size are broken by setting the appropriate pen values to 0.
310 The "threshold" parameter sets the relative size of the largest jump,
312 relative to the map range (as determined by a min/max operation). The
313 default size is 0.1.
314 NOTES
316 This almost never catches stuff near the apex of cylindrical maps,
318 because the anomalous vectors get arbitrarily small. This could be
319 improved somewhat by looking at individual runs of the pen and using a
320 relative length scale that is calibrated to the rest of each run. it
321 is probably not worth the computational overhead.
322 t_unit_sphere
324 $t = t_unit_sphere(<options>);
325 (Cartography) 3-D globe projection (conformal; authalic)
327 This is similar to the inverse of t_spherical, but the inverse
329 transform projects 3-D coordinates onto the unit sphere, yielding only
330 a 2-D (lon/lat) output. Similarly, the forward transform deprojects
331 2-D (lon/lat) coordinates onto the surface of a unit sphere.
332 The cartesian system has its Z axis pointing through the pole of the
334 (lon,lat) system, and its X axis pointing through the equator at the
335 prime meridian.
336 Unit sphere mapping is unusual in that it is both conformal and
338 authalic. That is possible because it properly embeds the sphere in
339 3-space, as a notional globe.
340 This is handy as an intermediate step in lots of transforms, as
342 Cartesian 3-space is cleaner to work with than spherical 2-space.
343 Higher dimensional indices are preserved, so that "rider" indices (such
345 as pen value) are propagated.
346 There is no oblique transform for t_unit_sphere, largely because it's
348 so easy to rotate the output using t_linear once it's out into
349 Cartesian space. In fact, the other projections implement oblique
350 transforms by wrapping t_linear with t_unit_sphere.
351 OPTIONS:
353 radius, Radius (default 1.0)
355 The radius of the sphere, for the inverse transform. (Radius is
356 ignored in the forward transform). Defaults to 1.0 so that the
357 resulting Cartesian coordinates are in units of "body radii".
358 t_rot_sphere
360 $t = t_rot_sphere({origin=>[<theta>,<phi>],roll=>[<roll>]});
361 (Cartography) Generate oblique projections
363 You feed in the origin in (theta,phi) and a roll angle, and you get
365 back out (theta', phi') coordinates. This is useful for making oblique
366 or transverse projections: just compose t_rot_sphere with your
367 favorite projection and you get an oblique one.
368 Most of the projections automagically compose themselves with
370 t_rot_sphere if you feed in an origin or roll angle.
371 t_rot_sphere converts the base plate caree projection (straight lon,
373 straight lat) to a Cassini projection.
374 OPTIONS
376 STANDARD POSITIONAL OPTIONS
378 t_orthographic
380 $t = t_orthographic(<options>);
381 (Cartography) Ortho. projection (azimuthal; perspective)
383 This is a perspective view as seen from infinite distance. You can
385 specify the sub-viewer point in (lon,lat) coordinates, and a rotation
386 angle of the map CW from (north=up). This is equivalent to specify
387 viewer roll angle CCW from (north=up).
388 t_orthographic is a convenience interface to t_unit_sphere -- it is
390 implemented as a composition of a t_unit_sphere call, a rotation, and a
391 slice.
392 [*] In the default case where the near hemisphere is mapped, the
394 inverse exists. There is no single inverse for the whole-sphere case,
395 so the inverse transform superimposes everything on a single
396 hemisphere. If you want an invertible 3-D transform, you want
397 t_unit_sphere.
398 OPTIONS
400 STANDARD POSITIONAL OPTIONS
402 m, mask, Mask, h, hemisphere, Hemisphere [default 'near']
403 The hemisphere to keep in the projection (see
404 PDL::Transform::Cartography).
405 NOTES
407 Alone of the various projections, this one does not use t_rot_sphere to
409 handle the standard options, because the cartesian coordinates of the
410 rotated sphere are already correctly projected -- t_rot_sphere would
411 put them back into (theta', phi') coordinates.
412 t_caree
414 $t = t_caree(<options>);
415 (Cartography) Plate Caree projection (cylindrical; equidistant)
417 This is the simple Plate Caree projection -- also called a "lat/lon
419 plot". The horizontal axis is theta; the vertical axis is phi. This
420 is a no-op if the angular unit is radians; it is a simple scale
421 otherwise.
422 OPTIONS
424 STANDARD POSITIONAL OPTIONS
426 s, std, standard, Standard (default 0)
427 The standard parallel where the transformation is conformal.
428 Conformality is achieved by shrinking of the horizontal scale to
429 match the vertical scale (which is correct everywhere).
430 t_mercator
432 $t = t_mercator(<options>);
433 (Cartography) Mercator projection (cylindrical; conformal)
435 This is perhaps the most famous of all map projections: meridians are
437 mapped to parallel vertical lines and parallels are unevenly spaced
438 horizontal lines. The poles are shifted to +/- infinity. The output
439 values are in units of globe-radii for easy conversion to kilometers;
440 hence the horizontal extent is -pi to pi.
441 You can get oblique Mercator projections by specifying the "origin" or
443 "roll" options; this is implemented via t_rot_sphere.
444 OPTIONS
446 STANDARD POSITIONAL OPTIONS
448 c, clip, Clip (default 75 [degrees])
449 The north/south clipping boundary of the transformation. Because
450 the poles are displaced to infinity, many applications require a
451 clipping boundary. The value is in whatever angular unit you set
452 with the standard 'units' option. The default roughly matches
453 interesting landforms on Earth. For no clipping at all, set b=>0.
454 For asymmetric clipping, use a 2-element list ref or piddle.
455 s, std, Standard (default 0)
457 This is the parallel at which the map has correct scale. The scale
458 is also correct at the parallel of opposite sign.
459 t_utm
461 $t = t_utm(<zone>,<options>);
462 (Cartography) Universal Transverse Mercator projection (cylindrical)
464 This is the internationally used UTM projection, with 2 subzones
466 (North/South). The UTM zones are parametrized individually, so if you
467 want a Zone 30 map you should use "t_utm(30)". By default you get the
468 northern subzone, so that locations in the southern hemisphere get
469 negative Y coordinates. If you select the southern subzone (with the
470 "subzone=>-1" option), you get offset southern UTM coordinates.
471 The 20-subzone military system is not yet supported. If/when it is
473 implemented, you will be able to enter "subzone=>[a-t]" to select a N/S
474 subzone.
475 Note that UTM is really a family of transverse Mercator projections
477 with different central meridia. Each zone properly extends for six
478 degrees of longitude on either side of its appropriate central
479 meridian, with Zone 1 being centered at -177 degrees longitude (177
480 west). Properly speaking, the zones only extend from 80 degrees south
481 to 84 degrees north; but this implementation lets you go all the way to
482 90 degrees. The default UTM coordinates are meters. The origin for
483 each zone is on the equator, at an easting of -500,000 meters.
484 The default output units are meters, assuming that you are wanting a
486 map of the Earth. This will break for bodies other than Earth (which
487 have different radii and hence different conversions between lat/lon
488 angle and meters).
489 The standard UTM projection has a slight reduction in scale at the
491 prime meridian of each zone: the transverse Mercator projection's
492 standard "parallels" are 180km e/w of the central meridian. However,
493 many Europeans prefer the "Gauss-Kruger" system, which is virtually
494 identical to UTM but with a normal tangent Mercator (standard parallel
495 on the prime meridian). To get this behavior, set "gk=>1".
496 Like the rest of the PDL::Transform::Cartography package, t_utm uses a
498 spherical datum rather than the "official" ellipsoidal datums for the
499 UTM system.
500 This implementation was derived from the rather nice description by
502 Denis J. Dean, located on the web at:
503 http://www.cnr.colostate.edu/class_info/nr502/lg3/datums_coordinates/utm.html
504 OPTIONS
506 STANDARD OPTIONS
508 (No positional options -- Origin and Roll are ignored)
509 ou, ounit, OutputUnit (default 'meters')
511 (This is likely to become a standard option in a future release) The
512 unit of the output map. By default, this is 'meters' for UTM, but
513 you may specify 'deg' or 'km' or even (heaven help us) 'miles' if
514 you prefer.
515 sz, subzone, SubZone (default 1)
517 Set this to -1 for the southern hemisphere subzone. Ultimately you
518 should be able to set it to a letter to get the corresponding
519 military subzone, but that's too much effort for now.
520 gk, gausskruger (default 0)
522 Set this to 1 to get the (European-style) tangent-plane Mercator
523 with standard parallel on the prime meridian. The default of 0
524 places the standard parallels 180km east/west of the prime meridian,
525 yielding better average scale across the zone. Setting gk=>1 makes
526 the scale exactly 1.0 at the central meridian, and >1.0 everywhere
527 else on the projection. The difference in scale is about 0.3%.
528 t_sin_lat
530 $t = t_sin_lat(<options>);
531 (Cartography) Cyl. equal-area projection (cyl.; authalic)
533 This projection is commonly used in solar Carrington plots; but not
535 much for terrestrial mapping.
536 OPTIONS
538 STANDARD POSITIONAL OPTIONS
540 s,std, Standard (default 0)
541 This is the parallel at which the map is conformal. It is also
542 conformal at the parallel of opposite sign. The conformality is
543 achieved by matched vertical stretching and horizontal squishing (to
544 achieve constant area).
545 t_sinusoidal
547 $t = t_sinusoidal(<options>);
548 (Cartography) Sinusoidal projection (authalic)
550 Sinusoidal projection preserves the latitude scale but scales longitude
552 according to sin(lat); in this respect it is the companion to
553 t_sin_lat, which is also authalic but preserves the longitude scale
554 instead.
555 OPTIONS
557 STANDARD POSITIONAL OPTIONS
559 t_conic
561 $t = t_conic(<options>)
562 (Cartography) Simple conic projection (conic; equidistant)
564 This is the simplest conic projection, with parallels mapped to
566 equidistant concentric circles. It is neither authalic nor conformal.
567 This transformation is also referred to as the "Modified Transverse
568 Mercator" projection in several maps of Alaska published by the USGS;
569 and the American State of New Mexico re-invented the projection in 1936
570 for an official map of that State.
571 OPTIONS
573 STANDARD POSITIONAL OPTIONS
575 s, std, Standard (default 29.5, 45.5)
576 The locations of the standard parallel(s) (where the cone intersects
577 the surface of the sphere). If you specify only one then the other
578 is taken to be the nearest pole. If you specify both of them to be
579 one pole then you get an equidistant azimuthal map. If you specify
580 both of them to be opposite and equidistant from the equator you get
581 a Plate Caree projection.
582 t_albers
584 $t = t_albers(<options>)
585 (Cartography) Albers conic projection (conic; authalic)
587 This is the standard projection used by the US Geological Survey for
589 sectionals of the 50 contiguous United States of America.
590 The projection reduces to the Lambert equal-area conic (infrequently
592 used and not to be confused with the Lambert conformal conic,
593 t_lambert!) if the pole is used as one of the two standard parallels.
594 Notionally, this is a conic projection onto a cone that intersects the
596 sphere at the two standard parallels; it works best when the two
597 parallels straddle the region of interest.
598 OPTIONS
600 STANDARD POSITIONAL OPTIONS
602 s, std, standard, Standard (default (29.5,45.5))
603 The locations of the standard parallel(s). If you specify only one
604 then the other is taken to be the nearest pole and a Lambert Equal-
605 Area Conic map results. If you specify both standard parallels to
606 be the same pole, then the projection reduces to the Lambert
607 Azimuthal Equal-Area map as aq special case. (Note that t_lambert
608 is Lambert's Conformal Conic, the most commonly used of Lambert's
609 projections.)
610 The default values for the standard parallels are those chosen by
612 Adams for maps of the lower 48 US states: (29.5,45.5). The USGS
613 recommends (55,65) for maps of Alaska and (8,18) for maps of Hawaii
614 -- these latter are chosen to also include the Canal Zone and
615 Philippine Islands farther south, which is why both of those
616 parallels are south of the Hawaiian islands.
617 The transformation reduces to the cylindrical equal-area (sin-lat)
619 transformation in the case where the standard parallels are opposite
620 and equidistant from the equator, and in fact this is implemented by
621 a call to t_sin_lat.
622 t_lambert
624 $t = t_lambert(<options>);
625 (Cartography) Lambert conic projection (conic; conformal)
627 Lambert conformal conic projection is widely used in aeronautical
629 charts and state base maps published by the USA's FAA and USGS. It's
630 especially useful for mid-latitude charts. In particular, straight
631 lines approximate (but are not exactly) great circle routes of up to ~2
632 radians.
633 The default standard parallels are 33 and 45 to match the USGS state
635 1:500,000 base maps of the United States. At scales of 1:500,000 and
636 larger, discrepancies between the spherical and ellipsoidal projections
637 become important; use care with this projection on spheres.
638 OPTIONS
640 STANDARD POSITIONAL OPTIONS
642 s, std, standard, Standard (default (33,45))
643 The locations of the standard parallel(s) for the conic projection.
644 The transform reduces to the Mercator projection in the case where
645 the standard parallels are opposite and equidistant from the
646 equator, and in fact this is implemented by a call to t_mercator.
647 c, clip, Clip (default [-75,75])
649 Because the transform is conformal, the distant pole is displaced to
650 infinity. Many applications require a clipping boundary. The value
651 is in whatever angular unit you set with the standard 'unit' option.
652 For consistency with t_mercator, clipping works the same way even
653 though in most cases only one pole needs it. Set this to 0 for no
654 clipping at all.
655 t_stereographic
657 $t = t_stereographic(<options>);
658 (Cartography) Stereographic projection (az.; conf.; persp.)
660 The stereographic projection is a true perspective (planar) projection
662 from a point on the spherical surface opposite the origin of the map.
663 OPTIONS
665 STANDARD POSITIONAL OPTIONS
667 c, clip, Clip (default 120)
668 This is the angular distance from the center to the edge of the
669 projected map. The default 120 degrees gives you most of the
670 opposite hemisphere but avoids the hugely distorted part near the
671 antipodes.
672 t_gnomonic
674 $t = t_gnomonic(<options>);
675 (Cartography) Gnomonic (focal-plane) projection (az.; persp.)
677 The gnomonic projection projects a hemisphere onto a tangent plane. It
679 is useful in cartography for the property that straight lines are great
680 circles; and it is useful in scientific imaging because it is the
681 projection generated by a simple optical system with a flat focal
682 plane.
683 OPTIONS
685 STANDARD POSITIONAL OPTIONS
687 c, clip, Clip (default 75)
688 This is the angular distance from the center to the edge of the
689 projected map. The default 75 degrees gives you most of the
690 hemisphere but avoids the hugely distorted part near the horizon.
691 t_az_eqd
693 $t = t_az_eqd(<options>);
694 (Cartography) Azimuthal equidistant projection (az.; equi.)
696 Basic azimuthal projection preserving length along radial lines from
698 the origin (meridians, in the original polar aspect). Hence, both
699 azimuth and distance are correct for journeys beginning at the origin.
700 Applied to the celestial sphere, this is the projection made by fisheye
702 lenses; it is also the projection into which "t_vertical" puts
703 perspective views.
704 The projected plane scale is normally taken to be planetary radii; this
706 is useful for cartographers but not so useful for scientific observers.
707 Setting the 't=>1' option causes the output scale to shift to camera
708 angular coordinates (the angular unit is determined by the standard
709 'Units' option; default is degrees).
710 OPTIONS
712 STANDARD POSITIONAL OPTIONS
714 c, clip, Clip (default 180 degrees)
715 The largest angle relative to the origin. Default is the whole
716 sphere.
717 t_az_eqa
719 $t = t_az_eqa(<options>);
720 (Cartography) Azimuthal equal-area projection (az.; auth.)
722 OPTIONS
724 STANDARD POSITIONAL OPTIONS
726 c, clip, Clip (default 180 degrees)
727 The largest angle relative to the origin. Default is the whole
728 sphere.
729 t_aitoff
731 "t_aitoff" in an alias for "t_hammer"
732 t_hammer
734 (Cartography) Hammer/Aitoff elliptical projection (az.; auth.)
735 The Hammer/Aitoff projection is often used to display the Celestial
737 sphere. It is mathematically related to the Lambert Azimuthal Equal-
738 Area projection (t_az_eqa), and maps the sphere to an ellipse of unit
739 eccentricity, with vertical radius sqrt(2) and horizontal radius of 2
740 sqrt(2).
741 OPTIONS
743 STANDARD POSITIONAL OPTIONS
745 t_zenithal
747 Vertical projections are also called "zenithal", and "t_zenithal" is an
748 alias for "t_vertical".
749 t_vertical
751 $t = t_vertical(<options>);
752 (Cartography) Vertical perspective projection (az.; persp.)
754 Vertical perspective projection is a generalization of gnomonic and
756 stereographic projection, and a special case of perspective projection.
757 It is a projection from the sphere onto a tangent plane from a point at
758 the camera location.
759 OPTIONS
761 STANDARD POSITIONAL OPTIONS
763 m, mask, Mask, h, hemisphere, Hemisphere [default 'near']
764 The hemisphere to keep in the projection (see
765 PDL::Transform::Cartography).
766 r0, R0, radius, d, dist, distance [default 2.0]
768 The altitude of the focal plane above the center of the sphere. The
769 default places the point of view one radius above the surface.
770 t, telescope, Telescope, cam, Camera (default '')
772 If this is set, then the central scale is in telescope or camera
773 angular units rather than in planetary radii. The angular units are
774 parsed as with the normal 'u' option for the lon/lat specification.
775 If you specify a non-string value (such as 1) then you get
776 telescope-frame radians, suitable for working on with other
777 transformations.
778 f, fish, fisheye (default '')
780 If this is set then the output is in azimuthal equidistant
781 coordinates instead of in tangent-plane coordinates. This is a
782 convenience function for '(t_az_eqd) x !(t_gnomonic) x
783 (t_vertical)'.
784 t_perspective
786 $t = t_perspective(<options>);
787 (Cartography) Arbitrary perspective projection
789 Perspective projection onto a focal plane from an arbitrary location
791 within or without the sphere, with an arbitrary central look direction,
792 and with correction for magnification within the optical system.
793 In the forward direction, t_perspective generates perspective views of
795 a sphere given (lon/lat) mapping or vector information. In the reverse
796 direction, t_perspective produces (lon/lat) maps from aerial or distant
797 photographs of spherical objects.
798 Viewpoints outside the sphere treat the sphere as opaque by default,
800 though you can use the 'm' option to specify either the near or far
801 surface (relative to the origin). Viewpoints below the surface treat
802 the sphere as transparent and undergo a mirror reversal for consistency
803 with projections that are special cases of the perspective projection
804 (e.g. t_gnomonic for r0=0 or t_stereographic for r0=-1).
805 Magnification correction handles the extra edge distortion due to
807 higher angles between the focal plane and focused rays within the
808 optical system of your camera. If you do not happen to know the
809 magnification of your camera, a simple rule of thumb is that the
810 magnification of a reflective telescope is roughly its focal length
811 (plate scale) divided by its physical length; and the magnification of
812 a compound refractive telescope is roughly twice its physical length
813 divided by its focal length. Simple optical systems with a single
814 optic have magnification = 1. Fisheye lenses have magnification < 1.
815 This transformation was derived by direct geometrical calculation
817 rather than being translated from Voxland & Snyder.
818 OPTIONS
820 STANDARD POSITIONAL OPTIONS
822 As always, the 'origin' field specifies the sub-camera point on the
823 sphere.
824 The 'roll' option is the roll angle about the sub-camera point, for
826 consistency with the other projectons.
827 p, ptg, pointing, Pointing (default (0,0,0))
829 The pointing direction, in (horiz. offset, vert. offset, roll) of
830 the camera relative to the center of the sphere. This is a
831 spherical coordinate system with the origin pointing directly at the
832 sphere and the pole pointing north in the pre-rolled coordinate
833 system set by the standard origin. It's most useful for space-based
834 images taken some distance from the body in question (e.g. images of
835 other planets or the Sun).
836 Be careful not to confuse 'p' (pointing) with 'P' (P angle, a
838 standard synonym for roll).
839 c, cam, camera, Camera (default undef)
841 Alternate way of specifying the camera pointing, using a spherical
842 coordinate system with poles at the zenith (positive) and nadir
843 (negative) -- this is useful for aerial photographs and such, where
844 the point of view is near the surface of the sphere. You specify
845 (azimuth from N, altitude from horizontal, roll from vertical=up).
846 If you specify pointing by this method, it overrides the 'pointing'
847 option, above. This coordinate system is most useful for aerial
848 photography or low-orbit work, where the nadir is not necessarily
849 the most interesting part of the scene.
850 r0, R0, radius, d, dist, distance [default 2.0]
852 The altitude of the point of view above the center of the sphere.
853 The default places the point of view 1 radius aboove the surface.
854 Do not confuse this with 'r', the standard origin roll angle!
855 Setting r0 < 1 gives a viewpoint inside the sphere. In that case,
856 the images are mirror-reversed to preserve the chiralty of the
857 perspective. Setting r0=0 gives gnomonic projections; setting r0=-1
858 gives stereographic projections. Setting r0 < -1 gives strange
859 results.
860 iu, im_unit, image_unit, Image_Unit (default 'degrees')
862 This is the angular units in which the viewing camera is calibrated
863 at the center of the image.
864 mag, magnification, Magnification (default 1.0)
866 This is the magnification factor applied to the optics -- it affects
867 the amount of tangent-plane distortion within the telescope. 1.0
868 yields the view from a simple optical system; higher values are
869 telescopic, while lower values are wide-angle (fisheye). Higher
870 magnification leads to higher angles within the optical system, and
871 more tangent-plane distortion at the edges of the image. The
872 magnification is applied to the incident angles themselves, rather
873 than to their tangents (simple two-element telescopes magnify
874 tan(theta) rather than theta itself); this is appropriate because
875 wide-field optics more often conform to the equidistant azimuthal
876 approximation than to the tangent plane approximation. If you need
877 more detailed control of the relationship between incident angle and
878 focal-plane position, use mag=1.0 and compose the transform with
879 something else to tweak the angles.
880 m, mask, Mask, h, hemisphere, Hemisphere [default 'near']
882 'hemisphere' is by analogy to other cartography methods although the
883 two regions to be selected are not really hemispheres.
884 f, fov, field_of_view, Field_Of_View [default 60 degrees]
886 The field of view of the telescope -- sets the crop radius on the
887 focal plane. If you pass in a scalar, you get a circular crop. If
888 you pass in a 2-element list ref, you get a rectilinear crop, with
889 the horizontal 'radius' and vertical 'radius' set separately.
890 EXAMPLES
892 Model a camera looking at the Sun through a 10x telescope from Earth
894 (~230 solar radii from the Sun), with an 0.5 degree field of view and a
895 solar P (roll) angle of 30 degrees, in February (sub-Earth solar
896 latitude is 7 degrees south). Convert a solar FITS image taken with
897 that camera to a FITS lon/lat map of the Sun with 20 pixels/degree
898 latitude:
899 # Define map output header (no need if you don't want a FITS output map)
901 $maphdr = {NAXIS1=>7200,NAXIS2=>3600, # Size of image
902 CTYPE1=>longitude,CTYPE2=>latitude, # Type of axes
903 CUNIT1=>deg,CUNIT2=>deg, # Unit of axes
904 CDELT1=>0.05,CDELT2=>0.05, # Scale of axes
905 CRPIX1=>3601,CRPIX2=>1801, # Center of map
906 CRVAL1=>0,CRVAL2=>0 # (lon,lat) of center
907 };
908 # Set up the perspective transformation, and apply it.
910 $t = t_perspective(r0=>229,fov=>0.5,mag=>10,P=>30,B=>-7);
911 $map = $im->map( $t , $maphdr );
912 Draw an aerial-view map of the Chesapeake Bay, as seen from a sounding
914 rocket at an altitude of 100km, looking NNE from ~200km south of
915 Washington (the radius of Earth is 6378 km; Washington D.C. is at
916 roughly 77W,38N). Superimpose a linear coastline map on a photographic
917 map.
918 $x = graticule(1,0.1)->glue(1,earth_coast());
920 $t = t_perspective(r0=>6478/6378.0,fov=>60,cam=>[22.5,-20],o=>[-77,36])
921 $w = pgwin(size=>[10,6],J=>1);
922 $w->fits_imag(earth_image()->map($t,[800,500],{m=>linear}));
923 $w->hold;
924 $w->lines($x->apply($t),{xt=>'Degrees',yt=>'Degrees'});
925 $w->release;
926 Model a 5x telescope looking at Betelgeuse with a 10 degree field of
928 view (since the telescope is looking at the Celestial sphere, r is 0
929 and this is just an expensive modified-gnomonic projection).
930 $t = t_perspective(r0=>0,fov=>10,mag=>5,o=>[88.79,7.41])
932perl v5.32.0 2020-09-17 Cartography(3)