Cartography(3) User Contributed Perl Documentation Cartography(3)NAMEPDL::Transform::Cartography - Useful cartographic projections
SYNOPSIS
# make a Mercator map of Earth
use PDL::Transform::Cartography;
$a = earth_coast();
$a = graticule(10,2)->glue(1,$a);
$t = t_mercator;
$w = pgwin(xs);
$w->lines($t->apply($a)->clean_lines());
DESCRIPTIONPDL::Transform::Cartography includes a variety of useful cartographic
and observing projections (mappings of the surface of a sphere),
including reprojected observer coordinates. See PDL::Transform for
more information about image transforms in general.
Cartographic transformations are used for projecting not just
terrestrial maps, but also any nearly spherical surface including the
Sun, the Celestial sphere, various moons and planets, distant stars,
etc. They also are useful for interpreting scientific images, which
are themselves generally projections of a sphere onto a flat focal
plane (e.g. the t_gnomonic projection).
Unless otherwise noted, all the transformations in this file convert
from (theta,phi) coordinates on the unit sphere (e.g. (lon,lat) on a
planet or (RA,dec) on the celestial sphere) into some sort of projected
coordinates, and have inverse transformations that convert back to
(theta,phi). This is equivalent to working from the equidistant
cylindrical (or "plate caree") projection, if you are a cartography
wonk.
The projected coordinates are generally in units of body radii
(radians), so that multiplying the output by the scale of the map
yields physical units that are correct wherever the scale is correct
for that projection. For example, areas should be correct everywhere
in the authalic projections; and linear scales are correct along
meridians in the equidistant projections and along the standard
parallels in all the projections.
The transformations that are authalic (equal-area), conformal (equal-
angle), azimuthal (circularly symmetric), or perspective (true
perspective on a focal plane from some viewpoint) are marked. The
first two categories are mutually exclusive for all but the "unit
sphere" 3-D projection.
Extra dimensions tacked on to each point to be transformed are, in
general, ignored. That is so that you can add on an extra index to
keep track of pen color. For example, earth_coast returns a 3x<n>
piddle containing (lon, lat, pen) at each list location. Transforming
the vector list retains the pen value as the first index after the
dimensional directions.
GENERAL NOTES ON CARTOGRAPHY
Unless otherwise noted, the transformations and miscellaneous
information in this section are taken from Snyder & Voxland 1989: "An
Album of Map Projections", US Geological Survey Professional Paper
1453, US Printing Office (Denver); and from Snyder 1987: "Map
Projections - A Working Manual", US Geological Survey Professional
Paper 1395, US Printing Office (Denver, USA). You can obtain your own
copy of both by contacting the U.S. Geological Survey, Federal Center,
Box 25425, Denver, CO 80225 USA.
The mathematics of cartography have a long history, and the details are
far trickier than the broad overview. For terrestrial (and, in
general, planetary) cartography, the best reference datum is not a
sphere but an oblate ellipsoid due to centrifugal force from the
planet's rotation. Furthermore, because all rocky planets, including
Earth, have randomly placed mass concentrations that affect the
gravitational field, the reference gravitational isosurface (sea level
on Earth) is even more complex than an ellipsoid and, in general,
different ellipsoids have been used for different locations at the same
time and for the same location at different times.
The transformations in this package use a spherical datum and hence
include global distortion at about the 0.5% level for terrestrial maps
(Earth's oblateness is ~1/300). This is roughly equal to the
dimensional precision of physical maps printed on paper (due to
stretching and warping of the paper) but is significant at larger
scales (e.g. for regional maps). If you need more precision than that,
you will want to implement and use the ellipsoidal transformations from
Snyder 1987 or another reference work on geodesy. A good name for that
package would be "...::Cartography::Geodetic".
GENERAL NOTES ON PERSPECTIVE AND SCIENTIFIC IMAGES
Cartographic transformations are useful for interpretation of
scientific images, as all cameras produce projections of the celestial
sphere onto the focal plane of the camera. A simple (single-element)
optical system with a planar focal plane generates gnomonic images --
that is to say, gnomonic projections of a portion of the celestial
sphere near the paraxial direction. This is the projection that most
consumer grade cameras produce.
Magnification in an optical system changes the angle of incidence of
the rays on the focal plane for a given angle of incidence at the
aperture. For example, a 10x telescope with a 2 degree field of view
exhibits the same gnomonic distortion as a simple optical system with a
20 degree field of view. Wide-angle optics typically have
magnification less than 1 ('fisheye lenses'), reducing the gnomonic
distortion considerably but introducing "equidistant azimuthal"
distortion -- there's no such thing as a free lunch!
Because many solar-system objects are spherical,
PDL::Transform::Cartography includes perspective projections for
producing maps of spherical bodies from perspective views. Those
projections are "t_vertical" and "t_perspective". They map between
(lat,lon) on the spherical body and planar projected coordinates at the
viewpoint. "t_vertical" is the vertical perspective projection given
by Snyder, but "t_perspective" is a fully general perspective
projection that also handles magnification correction.
TRANSVERSE & OBLIQUE PROJECTIONS; STANDARD OPTIONS
Oblique projections rotate the sphere (and graticule) to an arbitrary
angle before generating the projection; transverse projections rotate
the sphere exactly 90 degrees before generating the projection.
Most of the projections accept the following standard options, useful
for making transverse and oblique projection maps.
o, origin, Origin [default (0,0,0)]
The origin of the oblique map coordinate system, in (old-theta, old-
phi) coordinates.
r, roll, Roll [default 0.0]
The roll angle of the sphere about the origin, measured CW from (N =
up) for reasonable values of phi and CW from (S = up) for
unreasonable values of phi. This is equivalent to observer roll
angle CCW from the same direction.
u, unit, Unit [default 'degree']
This is the name of the angular unit to use in the lon/lat
coordinate system.
b, B
The "B" angle of the body -- used for extraterrestrial maps.
Setting this parameter is exactly equivalent to setting the phi
component of the origin, and in fact overrides it.
l,L
The longitude of the central meridian as observed -- used for
extraterrestrial maps. Setting this parameter is exactly equivalent
to setting the theta component of the origin, and in fact overrides
it.
p,P
The "P" (or position) angle of the body -- used for extraterrestrial
maps. This parameter is a synonym for the roll angle, above.
bad, Bad, missing, Missing [default nan]
This is the value that missing points get. Mainly useful for the
inverse transforms. (This should work fine if set to BAD, if you
have bad-value support compiled in). The default nan is asin(1.2),
calculated at load time.
EXAMPLES
Draw a Mercator map of the world on-screen:
$w = pgwin(xs);
$w->lines(earth_coast->apply(t_mercator)->clean_lines);
Here, "earth_coast()" returns a 3xn piddle containing (lon, lat, pen)
values for the included world coastal outline; "t_mercator" converts
the values to projected Mercator coordinates, and "clean_lines" breaks
lines that cross the 180th meridian.
Draw a Mercator map of the world, with lon/lat at 10 degree intervals:
$w = pgwin(xs)
$a = earth_coast()->glue(1,graticule(10,1));
$w->lines($a->apply(t_mercator)->clean_lines);
This works just the same as the first example, except that a map
graticule has been applied with interline spacing of 10 degrees lon/lat
and inter-vertex spacing of 1 degree (so that each meridian contains
181 points, and each parallel contains 361 points).
NOTES
Currently angular conversions are rather simpleminded. A list of
common conversions is present in the main constructor, which inserts a
conversion constant to radians into the {params} field of the new
transform. Something like Math::Convert::Units should be used instead
to generate the conversion constant.
A cleaner higher-level interface is probably needed (see the examples);
for example, earth_coast could return a graticule if asked, instead of
needing one to be glued on.
The class structure is somewhat messy because of the varying needs of
the different transformations. PDL::Transform::Cartography is a base
class that interprets the origin options and sets up the basic
machinery of the Transform. The conic projections have their own
subclass, PDL::Transform::Conic, that interprets the standard
parallels. Since the cylindrical and azimuthal projections are pretty
simple, they are not subclassed.
The perl 5.6.1 compiler is quite slow at adding new classes to the
structure, so it does not makes sense to subclass new transformations
merely for the sake of pedantry.
AUTHOR
Copyright 2002, Craig DeForest (deforest@boulder.swri.edu). This
module may be modified and distributed under the same terms as PDL
itself. The module comes with NO WARRANTY.
The included digital world map is derived from the 1987 CIA World Map,
translated to ASCII in 1988 by Joe Dellinger (geojoe@freeusp.org) and
simplified in 1995 by Kirk Johnson (tuna@indra.com) for the program
XEarth. The map comes with NO WARRANTY. An ASCII version of the map,
and a sample PDL function to read it, may be found in the Demos
subdirectory of the PDL source distribution.
FUNCTIONS
The module exports both transform constructors ('t_<foo>') and some
auxiliary functions (no leading 't_').
graticule
$lonlatp = graticule(<grid-spacing>,<line-segment-size>);
(Cartography) PDL constructor - generate a lat/lon grid.
Returns a grid of meridians and parallels as a list of vectors suitable
for sending to PDL::Graphics::PGPLOT::Window::lines for plotting. The
grid is in degrees in (theta, phi) coordinates -- this is (E lon, N
lat) for terrestrial grids or (RA, dec) for celestial ones. You must
then transform the graticule in the same way that you transform the
map.
You can attach the graticule to a vector map using the syntax:
$out = graticule(10,2)->glue(1,$map);
In array context you get back a 2-element list containing a piddle of
the (theta,phi) pairs and a piddle of the pen values (1 or 0) suitable
for calling PDL::Graphics::PGPLOT::Window::lines. In scalar context
the two elements are combined into a single piddle.
The pen values associated with the graticule are negative, which will
cause PDL::Graphics::PGPLOT::Window::lines to plot them as hairlines.
earth_coast
$a = earth_coast()
(Cartography) PDL constructor - coastline map of Earth
Returns a vector coastline map based on the 1987 CIA World Coastline
database (see author information). The vector coastline data are in
plate caree format so they can be converted to other projections via
the apply method and cartographic transforms, and are suitable for
plotting with the lines method in the PGPLOT output library: the first
dimension is (X,Y,pen) with breaks having a pen value of 0 and
hairlines having negative pen values. The second dimension threads
over all the points in the data set.
The vector map includes lines that pass through the antipodean
meridian, so if you want to plot it without reprojecting, you should
run it through clean_lines first:
$w = pgwin();
$w->lines(earth_coast->clean_lines); # plot plate caree map of world
$w->lines(earth_coast->apply(t_gnomonic))# plot gnomonic map of world
"earth_coast" is just a quick-and-dirty way of loading the file
"earth_coast.vec.fits" that is part of the normal installation tree.
earth_image
$rgb = earth_image()
(Cartography) PDL constructor - RGB pixel map of Earth
Returns an RGB image of Earth based on data from the MODIS instrument
on the NASA EOS/Terra satellite. (You can get a full-resolution image
from <http://earthobservatory.nasa.gov/Newsroom/BlueMarble/>). The
image is a plate caree map, so you can convert it to other projections
via the map method and cartographic transforms.
This is just a quick-and-dirty way of loading the earth-image files
that are distributed along with PDL.
clean_lines
$a = clean_lines(t_mercator->apply(scalar(earth_coast())));
$a = clean_lines($line_pen, [threshold]);
$a = $lines->clean_lines;
(Cartography) PDL method - remove projection irregularities
"clean_lines" massages vector data to remove jumps due to singularities
in the transform.
In the first (scalar) form, $line_pen contains both (X,Y) points and
pen values suitable to be fed to lines: in the second (list) form,
$lines contains the (X,Y) points and $pen contains the pen values.
"clean_lines" assumes that all the outline polylines are local -- that
is to say, there are no large jumps. Any jumps larger than a threshold
size are broken by setting the appropriate pen values to 0.
The "threshold" parameter sets the relative size of the largest jump,
relative to the map range (as determined by a min/max operation). The
default size is 0.1.
NOTES
This almost never catches stuff near the apex of cylindrical maps,
because the anomalous vectors get arbitrarily small. This could be
improved somewhat by looking at individual runs of the pen and using a
relative length scale that is calibrated to the rest of each run. it
is probably not worth the computational overhead.
t_unit_sphere
$t = t_unit_sphere(<options>);
(Cartography) 3-D globe projection (conformal; authalic)
This is similar to the inverse of t_spherical, but the inverse
transform projects 3-D coordinates onto the unit sphere, yielding only
a 2-D (lon/lat) output. Similarly, the forward transform deprojects
2-D (lon/lat) coordinates onto the surface of a unit sphere.
The cartesian system has its Z axis pointing through the pole of the
(lon,lat) system, and its X axis pointing through the equator at the
prime meridian.
Unit sphere mapping is unusual in that it is both conformal and
authalic. That is possible because it properly embeds the sphere in
3-space, as a notional globe.
This is handy as an intermediate step in lots of transforms, as
Cartesian 3-space is cleaner to work with than spherical 2-space.
Higher dimensional indices are preserved, so that "rider" indices (such
as pen value) are propagated.
There is no oblique transform for t_unit_sphere, largely because it's
so easy to rotate the output using t_linear once it's out into
Cartesian space. In fact, the other projections implement oblique
transforms by wrapping t_linear with t_unit_sphere.
OPTIONS:
radius, Radius (default 1.0)
The radius of the sphere, for the inverse transform. (Radius is
ignored in the forward transform). Defaults to 1.0 so that the
resulting Cartesian coordinates are in units of "body radii".
t_rot_sphere
$t = t_rot_sphere({origin=>[<theta>,<phi>],roll=>[<roll>]});
(Cartography) Generate oblique projections
You feed in the origin in (theta,phi) and a roll angle, and you get
back out (theta', phi') coordinates. This is useful for making oblique
or transverse projections: just compose t_rot_sphere with your
favorite projection and you get an oblique one.
Most of the projections automagically compose themselves with
t_rot_sphere if you feed in an origin or roll angle.
t_rot_sphere converts the base plate caree projection (straight lon,
straight lat) to a Cassini projection.
OPTIONS
STANDARD POSITIONAL OPTIONS
t_orthographic
$t = t_orthographic(<options>);
(Cartography) Ortho. projection (azimuthal; perspective)
This is a perspective view as seen from infinite distance. You can
specify the sub-viewer point in (lon,lat) coordinates, and a rotation
angle of the map CW from (north=up). This is equivalent to specify
viewer roll angle CCW from (north=up).
t_orthographic is a convenience interface to t_unit_sphere -- it is
implemented as a composition of a t_unit_sphere call, a rotation, and a
slice.
[*] In the default case where the near hemisphere is mapped, the
inverse exists. There is no single inverse for the whole-sphere case,
so the inverse transform superimposes everything on a single
hemisphere. If you want an invertible 3-D transform, you want
t_unit_sphere.
OPTIONS
STANDARD POSITIONAL OPTIONS
m, mask, Mask, h, hemisphere, Hemisphere [default 'near']
The hemisphere to keep in the projection (see
PDL::Transform::Cartography).
NOTES
Alone of the various projections, this one does not use t_rot_sphere to
handle the standard options, because the cartesian coordinates of the
rotated sphere are already correctly projected -- t_rot_sphere would
put them back into (theta', phi') coordinates.
t_caree
$t = t_caree(<options>);
(Cartography) Plate Caree projection (cylindrical; equidistant)
This is the simple Plate Caree projection -- also called a "lat/lon
plot". The horizontal axis is theta; the vertical axis is phi. This
is a no-op if the angular unit is radians; it is a simple scale
otherwise.
OPTIONS
STANDARD POSITIONAL OPTIONS
s, std, standard, Standard (default 0)
The standard parallel where the transformation is conformal.
Conformality is achieved by shrinking of the horizontal scale to
match the vertical scale (which is correct everywhere).
t_mercator
$t = t_mercator(<options>);
(Cartography) Mercator projection (cylindrical; conformal)
This is perhaps the most famous of all map projections: meridians are
mapped to parallel vertical lines and parallels are unevenly spaced
horizontal lines. The poles are shifted to +/- infinity. The output
values are in units of globe-radii for easy conversion to kilometers;
hence the horizontal extent is -pi to pi.
You can get oblique Mercator projections by specifying the "origin" or
"roll" options; this is implemented via t_rot_sphere.
OPTIONS
STANDARD POSITIONAL OPTIONS
c, clip, Clip (default 75 [degrees])
The north/south clipping boundary of the transformation. Because
the poles are displaced to infinity, many applications require a
clipping boundary. The value is in whatever angular unit you set
with the standard 'units' option. The default roughly matches
interesting landforms on Earth. For no clipping at all, set b=>0.
For asymmetric clipping, use a 2-element list ref or piddle.
s, std, Standard (default 0)
This is the parallel at which the map has correct scale. The scale
is also correct at the parallel of opposite sign.
t_utm
$t = t_utm(<zone>,<options>);
(Cartography) Universal Transverse Mercator projection (cylindrical)
This is the internationally used UTM projection, with 2 subzones
(North/South). The UTM zones are parametrized individually, so if you
want a Zone 30 map you should use "t_utm(30)". By default you get the
northern subzone, so that locations in the southern hemisphere get
negative Y coordinates. If you select the southern subzone (with the
"subzone=>-1" option), you get offset southern UTM coordinates.
The 20-subzone military system is not yet supported. If/when it is
implemented, you will be able to enter "subzone=>[a-t]" to select a N/S
subzone.
Note that UTM is really a family of transverse Mercator projections
with different central meridia. Each zone properly extends for six
degrees of longitude on either side of its appropriate central
meridian, with Zone 1 being centered at -177 degrees longitude (177
west). Properly speaking, the zones only extend from 80 degrees south
to 84 degrees north; but this implementation lets you go all the way to
90 degrees. The default UTM coordinates are meters. The origin for
each zone is on the equator, at an easting of -500,000 meters.
The default output units are meters, assuming that you are wanting a
map of the Earth. This will break for bodies other than Earth (which
have different radii and hence different conversions between lat/lon
angle and meters).
The standard UTM projection has a slight reduction in scale at the
prime meridian of each zone: the transverse Mercator projection's
standard "parallels" are 180km e/w of the central meridian. However,
many Europeans prefer the "Gauss-Kruger" system, which is virtually
identical to UTM but with a normal tangent Mercator (standard parallel
on the prime meridian). To get this behavior, set "gk=>1".
Like the rest of the PDL::Transform::Cartography package, t_utm uses a
spherical datum rather than the "official" ellipsoidal datums for the
UTM system.
This implementation was derived from the rather nice description by
Denis J. Dean, located on the web at:
http://www.cnr.colostate.edu/class_info/nr502/lg3/datums_coordinates/utm.html
OPTIONS
STANDARD OPTIONS
(No positional options -- Origin and Roll are ignored)
ou, ounit, OutputUnit (default 'meters')
(This is likely to become a standard option in a future release) The
unit of the output map. By default, this is 'meters' for UTM, but
you may specify 'deg' or 'km' or even (heaven help us) 'miles' if
you prefer.
sz, subzone, SubZone (default 1)
Set this to -1 for the southern hemisphere subzone. Ultimately you
should be able to set it to a letter to get the corresponding
military subzone, but that's too much effort for now.
gk, gausskruger (default 0)
Set this to 1 to get the (European-style) tangent-plane Mercator
with standard parallel on the prime meridian. The default of 0
places the standard parallels 180km east/west of the prime meridian,
yielding better average scale across the zone. Setting gk=>1 makes
the scale exactly 1.0 at the central meridian, and >1.0 everywhere
else on the projection. The difference in scale is about 0.3%.
t_sin_lat
$t = t_sin_lat(<options>);
(Cartography) Cyl. equal-area projection (cyl.; authalic)
This projection is commonly used in solar Carrington plots; but not
much for terrestrial mapping.
OPTIONS
STANDARD POSITIONAL OPTIONS
s,std, Standard (default 0)
This is the parallel at which the map is conformal. It is also
conformal at the parallel of opposite sign. The conformality is
achieved by matched vertical stretching and horizontal squishing (to
achieve constant area).
t_sinusoidal
$t = t_sinusoidal(<options>);
(Cartography) Sinusoidal projection (authalic)
Sinusoidal projection preserves the latitude scale but scales longitude
according to sin(lat); in this respect it is the companion to
t_sin_lat, which is also authalic but preserves the longitude scale
instead.
OPTIONS
STANDARD POSITIONAL OPTIONS
t_conic
$t = t_conic(<options>)
(Cartography) Simple conic projection (conic; equidistant)
This is the simplest conic projection, with parallels mapped to
equidistant concentric circles. It is neither authalic nor conformal.
This transformation is also referred to as the "Modified Transverse
Mercator" projection in several maps of Alaska published by the USGS;
and the American State of New Mexico re-invented the projection in 1936
for an official map of that State.
OPTIONS
STANDARD POSITIONAL OPTIONS
s, std, Standard (default 29.5, 45.5)
The locations of the standard parallel(s) (where the cone intersects
the surface of the sphere). If you specify only one then the other
is taken to be the nearest pole. If you specify both of them to be
one pole then you get an equidistant azimuthal map. If you specify
both of them to be opposite and equidistant from the equator you get
a Plate Caree projection.
t_albers
$t = t_albers(<options>)
(Cartography) Albers conic projection (conic; authalic)
This is the standard projection used by the US Geological Survey for
sectionals of the 50 contiguous United States of America.
The projection reduces to the Lambert equal-area conic (infrequently
used and not to be confused with the Lambert conformal conic,
t_lambert!) if the pole is used as one of the two standard parallels.
Notionally, this is a conic projection onto a cone that intersects the
sphere at the two standard parallels; it works best when the two
parallels straddle the region of interest.
OPTIONS
STANDARD POSITIONAL OPTIONS
s, std, standard, Standard (default (29.5,45.5))
The locations of the standard parallel(s). If you specify only one
then the other is taken to be the nearest pole and a Lambert Equal-
Area Conic map results. If you specify both standard parallels to
be the same pole, then the projection reduces to the Lambert
Azimuthal Equal-Area map as aq special case. (Note that t_lambert
is Lambert's Conformal Conic, the most commonly used of Lambert's
projections.)
The default values for the standard parallels are those chosen by
Adams for maps of the lower 48 US states: (29.5,45.5). The USGS
recommends (55,65) for maps of Alaska and (8,18) for maps of Hawaii
-- these latter are chosen to also include the Canal Zone and
Philippine Islands farther south, which is why both of those
parallels are south of the Hawaiian islands.
The transformation reduces to the cylindrical equal-area (sin-lat)
transformation in the case where the standard parallels are opposite
and equidistant from the equator, and in fact this is implemented by
a call to t_sin_lat.
t_lambert
$t = t_lambert(<options>);
(Cartography) Lambert conic projection (conic; conformal)
Lambert conformal conic projection is widely used in aeronautical
charts and state base maps published by the USA's FAA and USGS. It's
especially useful for mid-latitude charts. In particular, straight
lines approximate (but are not exactly) great circle routes of up to ~2
radians.
The default standard parallels are 33 and 45 to match the USGS state
1:500,000 base maps of the United States. At scales of 1:500,000 and
larger, discrepancies between the spherical and ellipsoidal projections
become important; use care with this projection on spheres.
OPTIONS
STANDARD POSITIONAL OPTIONS
s, std, standard, Standard (default (33,45))
The locations of the standard parallel(s) for the conic projection.
The transform reduces to the Mercator projection in the case where
the standard parallels are opposite and equidistant from the
equator, and in fact this is implemented by a call to t_mercator.
c, clip, Clip (default [-75,75])
Because the transform is conformal, the distant pole is displaced to
infinity. Many applications require a clipping boundary. The value
is in whatever angular unit you set with the standard 'unit' option.
For consistency with t_mercator, clipping works the same way even
though in most cases only one pole needs it. Set this to 0 for no
clipping at all.
t_stereographic
$t = t_stereographic(<options>);
(Cartography) Stereographic projection (az.; conf.; persp.)
The stereographic projection is a true perspective (planar) projection
from a point on the spherical surface opposite the origin of the map.
OPTIONS
STANDARD POSITIONAL OPTIONS
c, clip, Clip (default 120)
This is the angular distance from the center to the edge of the
projected map. The default 120 degrees gives you most of the
opposite hemisphere but avoids the hugely distorted part near the
antipodes.
t_gnomonic
$t = t_gnomonic(<options>);
(Cartography) Gnomonic (focal-plane) projection (az.; persp.)
The gnomonic projection projects a hemisphere onto a tangent plane. It
is useful in cartography for the property that straight lines are great
circles; and it is useful in scientific imaging because it is the
projection generated by a simple optical system with a flat focal
plane.
OPTIONS
STANDARD POSITIONAL OPTIONS
c, clip, Clip (default 75)
This is the angular distance from the center to the edge of the
projected map. The default 75 degrees gives you most of the
hemisphere but avoids the hugely distorted part near the horizon.
t_az_eqd
$t = t_az_eqd(<options>);
(Cartography) Azimuthal equidistant projection (az.; equi.)
Basic azimuthal projection preserving length along radial lines from
the origin (meridians, in the original polar aspect). Hence, both
azimuth and distance are correct for journeys beginning at the origin.
Applied to the celestial sphere, this is the projection made by fisheye
lenses; it is also the projection into which "t_vertical" puts
perspective views.
The projected plane scale is normally taken to be planetary radii; this
is useful for cartographers but not so useful for scientific observers.
Setting the 't=>1' option causes the output scale to shift to camera
angular coordinates (the angular unit is determined by the standard
'Units' option; default is degrees).
OPTIONS
STANDARD POSITIONAL OPTIONS
c, clip, Clip (default 180 degrees)
The largest angle relative to the origin. Default is the whole
sphere.
t_az_eqa
$t = t_az_eqa(<options>);
(Cartography) Azimuthal equal-area projection (az.; auth.)
OPTIONS
STANDARD POSITIONAL OPTIONS
c, clip, Clip (default 180 degrees)
The largest angle relative to the origin. Default is the whole
sphere.
t_aitoff
t_hammer
(Cartography) Hammer/Aitoff elliptical projection (az.; auth.)
The Hammer/Aitoff projection is often used to display the Celestial
sphere. It is mathematically related to the Lambert Azimuthal Equal-
Area projection (t_az_eqa), and maps the sphere to an ellipse of unit
eccentricity, with vertical radius sqrt(2) and horizontal radius of 2
sqrt(2).
OPTIONS
STANDARD POSITIONAL OPTIONS
t_vertical
$t = t_vertical(<options>);
(Cartography) Vertical perspective projection (az.; persp.)
Vertical perspective projection is a generalization of gnomonic and
stereographic projection, and a special case of perspective projection.
It is a projection from the sphere onto a focal plane at the camera
location.
OPTIONS
STANDARD POSITIONAL OPTIONS
m, mask, Mask, h, hemisphere, Hemisphere [default 'near']
The hemisphere to keep in the projection (see
PDL::Transform::Cartography).
r0, R0, radius, d, dist, distance [default 2.0]
The altitude of the focal plane above the center of the sphere. The
default places the point of view one radius above the surface.
t, telescope, Telescope, cam, Camera (default '')
If this is set, then the central scale is in telescope or camera
angular units rather than in planetary radii. The angular units are
parsed as with the normal 'u' option for the lon/lat specification.
If you specify a non-string value (such as 1) then you get
telescope-frame radians, suitable for working on with other
transformations.
f, fish, fisheye (default '')
If this is set then the output is in azimuthal equidistant
coordinates instead of in tangent-plane coordinates. This is a
convenience function for '(t_az_eqd) x !(t_gnomonic) x
(t_vertical)'.
t_perspective
$t = t_perspective(<options>);
(Cartography) Arbitrary perspective projection
Perspective projection onto a focal plane from an arbitrary location
within or without the sphere, with an arbitary central look direction,
and with correction for magnification within the optical system.
In the forward direction, t_perspective generates perspective views of
a sphere given (lon/lat) mapping or vector information. In the reverse
direction, t_perspective produces (lon/lat) maps from aerial or distant
photographs of spherical objects.
Viewpoints outside the sphere treat the sphere as opaque by default,
though you can use the 'm' option to specify either the near or far
surface (relative to the origin). Viewpoints below the surface treat
the sphere as transparent and undergo a mirror reversal for consistency
with projections that are special cases of the perspective projection
(e.g. t_gnomonic for r0=0 or t_stereographic for r0=-1).
Magnification correction handles the extra edge distortion due to
higher angles between the focal plane and focused rays within the
optical system of your camera. If you do not happen to know the
magnification of your camera, a simple rule of thumb is that the
magnification of a reflective telescope is roughly its focal length
(plate scale) divided by its physical length; and the magnification of
a compound refractive telescope is roughly twice its physical length
divided by its focal length. Simple optical sytems with a single optic
have magnification = 1. Fisheye lenses have magnification < 1.
This transformation was derived by direct geometrical calculation
rather than being translated from Voxland & Snyder.
OPTIONS
STANDARD POSITIONAL OPTIONS
As always, the 'origin' field specifies the sub-camera point on the
sphere.
The 'roll' option is the roll angle about the sub-camera point, for
consistency with the other projectons.
p, ptg, pointing, Pointing (default (0,0,0))
The pointing direction, in (horiz. offset, vert. offset, roll) of
the camera relative to the center of the sphere. This is a
spherical coordinate system with the origin pointing directly at the
sphere and the pole pointing north in the pre-rolled coordinate
system set by the standard origin. It's most useful for space-based
images taken some distance from the body in question (e.g. images of
other planets or the Sun).
Be careful not to confuse 'p' (pointing) with 'P' (P angle, a
standard synonym for roll).
c, cam, camera, Camera (default undef)
Alternate way of specifying the camera pointing, using a spherical
coordinate system with poles at the zenith (positive) and nadir
(negative) -- this is useful for aerial photographs and such, where
the point of view is near the surface of the sphere. You specify
(azimuth from N, altitude from horizontal, roll from vertical=up).
If you specify pointing by this method, it overrides the 'pointing'
option, above. This coordinate system is most useful for aerial
photography or low-orbit work, where the nadir is not necessarily
the most interesting part of the scene.
r0, R0, radius, d, dist, distance [default 2.0]
The altitude of the point of view above the center of the sphere.
The default places the point of view 1 radius aboove the surface.
Do not confuse this with 'r', the standard origin roll angle!
Setting r0 < 1 gives a viewpoint inside the sphere. In that case,
the images are mirror-reversed to preserve the chiralty of the
perspective. Setting r0=0 gives gnomonic projections; setting r0=-1
gives stereographic projections. Setting r0 < -1 gives strange
results.
iu, im_unit, image_unit, Image_Unit (default 'degrees')
This is the angular units in which the viewing camera is calibrated
at the center of the image.
mag, magnification, Magnification (default 1.0)
This is the magnification factor applied to the optics -- it affects
the amount of tangent-plane distortion within the telescope. 1.0
yields the view from a simple optical system; higher values are
telescopic, while lower values are wide-angle (fisheye). Higher
magnification leads to higher angles within the optical system, and
more tangent-plane distortion at the edges of the image. The
magnification is applied to the incident angles themselves, rather
than to their tangents (simple two-element telescopes magnify
tan(theta) rather than theta itself); this is appropriate because
wide-field optics more often conform to the equidistant azimuthal
approximation than to the tangent plane approximation. If you need
more detailed control of the relationship between incident angle and
focal-plane position, use mag=1.0 and compose the transform with
something else to tweak the angles.
m, mask, Mask, h, hemisphere, Hemisphere [default 'near']
'hemisphere' is by analogy to other cartography methods although the
two regions to be selected are not really hemispheres.
f, fov, field_of_view, Field_Of_View [default 60 degrees]
The field of view of the telescope -- sets the crop radius on the
focal plane. If you pass in a scalar, you get a circular crop. If
you pass in a 2-element list ref, you get a rectilinear crop, with
the horizontal 'radius' and vertical 'radius' set separately.
EXAMPLES
Model a camera looking at the Sun through a 10x telescope from Earth
(~230 solar radii from the Sun), with an 0.5 degree field of view and a
solar P (roll) angle of 30 degrees, in February (sub-Earth solar
latitude is 7 degrees south). Convert a solar FITS image taken with
that camera to a FITS lon/lat map of the Sun with 20 pixels/degree
latitude:
# Define map output header (no need if you don't want a FITS output map)
$maphdr = {NAXIS1=>7200,NAXIS2=>3600, # Size of image
CTYPE1=>longitude,CTYPE2=>latitude, # Type of axes
CUNIT1=>deg,CUNIT2=>deg, # Unit of axes
CDELT1=>0.05,CDELT2=>0.05, # Scale of axes
CRPIX1=>3601,CRPIX2=>1801, # Center of map
CRVAL1=>0,CRVAL2=>0 # (lon,lat) of center
};
# Set up the perspective transformation, and apply it.
$t = t_perspective(r0=>229,fov=>0.5,mag=>10,P=>30,B=>-7);
$map = $im->map( $t , $maphdr );
Draw an aerial-view map of the Chesapeake Bay, as seen from a sounding
rocket at an altitude of 100km, looking NNE from ~200km south of
Washington (the radius of Earth is 6378 km; Washington D.C. is at
roughly 77W,38N). Superimpose a linear coastline map on a photographic
map.
$a = graticule(1,0.1)->glue(1,earth_coast());
$t = t_perspective(r0=>6478/6378.0,fov=>60,cam=>[22.5,-20],o=>[-77,36])
$w = pgwin(size=>[10,6],J=>1);
$w->fits_imag(earth_image()->map($t,[800,500],{m=>linear}));
$w->hold;
$w->lines($a->apply($t),{xt=>'Degrees',yt=>'Degrees'});
$w->release;
Model a 5x telescope looking at Betelgeuse with a 10 degree field of
view (since the telescope is looking at the Celestial sphere, r is 0
and this is just an expensive modified-gnomonic projection).
$t = t_perspective(r0=>0,fov=>10,mag=>5,o=>[88.79,7.41])
perl v5.10.0 2004-07-21 Cartography(3)
