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With azimuthal projections, the UV plane is tangent to the globe. The point of tangency is projected onto the center of the plane and its latitude and longitude are P0lat and P0lon, respectively. Rot is the angle between North and the v-axis.
Important characteristics of azimuthal maps include the fact that directions or azimuths are correct from the center of the projection to any other point, and great circles through the center are projected to straight lines on the plane.
The IDL mapping package includes the following azimuthal projections: orthographic, stereographic, gnomonic, azimuthal equidistant, Aitoff, Lambert's azimuthal equal area, Hammer-Aitoff, and satellite.
The orthographic projection was known by the Egyptians and Greeks 2000 years ago. This projection looks like a globe because it is a perspective projection from infinite distance. As such, it maps one hemisphere of the globe into the UV plane. Distortions are greatest along the rim of the hemisphere where distances and land masses are compressed.
The following statements are used to produce an orthographic projection centered over Eastern Spain at a scale of 70 million to 1:
MAP_SET, /ORTHOGRAPHIC, 40, 0, SCALE=70e6, /CONTINENTS, $ /GRID, LONDEL=15, LATDEL=15, $ TITLE = 'Oblique Orthographic'
The output of these statements is shown in the following figure.
The stereographic projection is a true perspective projection with the globe being projected onto the UV plane from the point P on the globe diametrically opposite to the point of tangency. The whole globe except P is mapped onto the UV plane. There is great distortion for regions close to P, since P maps to infinity.
The stereographic projection is the only known perspective projection that is also conformal. It is frequently used for polar maps. For example, a stereographic view of the north pole has the south pole as its point of perspective.
The following statement uses the stereographic projection to draw the hemisphere centered on the equator at longitude -105 degrees and produces an equatorial stereographic map:
MAP_SET, /STEREO, 0, -105, /ISOTROPIC, $ /GRID, LATDEL = 20, LONDEL = 20, /HORIZON, /CONTINENT, $ TITLE = 'Equatorial Stereographic'
The output of this statement is shown in the following figure:
Since the LATDEL and LONDEL keywords are set to 20, parallels and meridians are spaced 20 degrees apart. The GRID and CONTINENT keywords signal that the grid and continents should be drawn.
The gnomonic projection (also called Central or Gnomic) projects all great circles to straight lines. The gnomonic projection is the perspective, azimuthal projection with point of perspective at the center of the globe. Hence, with the gnomonic projection, the interior of a hemispherical region of the globe is projected to the UV plane with the rim of the hemisphere going to infinity. Except at the center, there is great distortion of shape, area, and scale. The default clipping region for the gnomonic projection is a circle with a radius of 60 degrees at the center of projection.
MAP_SET, /GNOMIC, 40, -105, LIMIT = [20, -130, 70, -70], $ /ISOTROPIC, /GRID, /CONTINENT, $ TITLE = 'Oblique Gnomonic'
This projection is centered around the point at latitude 40 degrees and longitude -105 degrees. The region on the globe that is mapped lies between 20 degrees and 70 degrees of latitude and -130 degrees and -70 degrees of longitude.
The output of these statements is shown in the following figure:
The azimuthal equidistant projection is also not a true perspective projection, because it preserves correctly the distances between the tangent point and all other points on the globe. The point P opposite the tangent point is mapped to a circle on the UV plane, and hence, the whole globe is mapped to the plane. There is infinite distortion close to the outer rim of the map, which is the circular image of P.
If the keyword LIMIT is not set, the whole globe is mapped to the UV plane.
MAP_SET, /AZIMUTHAL, /ISOTROPIC, -90, $ /GRID, LONDEL=20, LATDEL=20, /CONTINENT, $ /HORIZON, TITLE = 'Polar Azimuthal'
It is centered at the South Pole and shows the entire globe.
The output of these statements is shown in the following figure:
The Aitoff projection modifies the equatorial aspect of one hemisphere of the azimuthal equidistant projection, described above. Lines parallel to the equator are stretched horizontally and meridian values are doubled, thereby displaying the world as an ellipse with axes in a 2:1 ratio. Both the equator and the central meridian are represented at true scale; however, distances measured between the point of tangency and any other point on the map are no longer true to scale.
An Aitoff projection centered on the international dateline can be produced by the command:
MAP_SET, 0, 180, /Aitoff, /GRID, /CONTINENTS, /ISOTROPIC, $ TITLE= 'Aitoff Projection'
The output of these statements is shown in the following figure:
Lambert's equal area projection adjusts projected distances in order to preserve area. Hence, it is not a true perspective projection. Like the stereographic projection, it maps to infinity the point P diametrically opposite the point of tangency. Note also that to preserve area, distances between points become more contracted as the points become closer to P. Lambert's equal area projection has less overall scale variation than the other azimuthal projections.
MAP_SET, /LAMBERT, 90, 0, -105, /ISOTROPIC, $
/GRID, LATDEL=20, LONDEL=20, $
/CONTINENTS, E_CONTINENTS={FILL:1}, /HORIZON, $
TITLE = 'Polar Lambert'
The output of these statements is shown in the following figure:
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Although the Hammer-Aitoff projection is not truly azimuthal, it is included in this section because it is derived from the equatorial aspect of Lambert's equal area projection limited to a hemisphere (in the same way Aitoff's projection is derived from the equatorial aspect of the azimuthal equidistant projection). In this derivation, the hemisphere is represented inside an ellipse with the rest of the world in the lunes of the ellipse.
Because the Hammer-Aitoff projection produces an equal area map of the entire globe, it is useful for visual representations of geographically related statistical data and distributions. Astronomers use this projection to show the entire celestial sphere on one map in a way that accurately depicts the relative distribution of the stars in different regions of the sky.
A Hammer-Aitoff projection centered on the international dateline can be produced by the command:
MAP_SET, 0, 180, /HAMMER, /GRID, /CONTINENTS, /ISOTROPIC, $ /HORIZON, TITLE= 'Hammer-Aitoff Projection'
The output of these statements is shown in the following figure:
The satellite projection, also called the General Perspective projection, simulates a view of the globe as seen from a camera in space. If the camera faces the center of the globe, the projection is called a Vertical Perspective projection (note that the orthographic, stereographic, and gnomonic projections are special cases of this projection), otherwise the projection is called a Tilted Perspective projection.
The globe is viewed from a point in space, with the viewing plane touching the surface of the globe at the point directly beneath the satellite (the sub-satellite point). If the projection plane is perpendicular to the line connecting the point of projection and the center of the globe, a Vertical Perspective projection results. Otherwise, the projection plane is horizontally turned G degrees clockwise from the north, then tilted w degrees downward from horizontal.
For the satellite projection, P0Lat and P0Lon represent the latitude and longitude of the sub-satellite point. Three additional parameters, P, Omega, and Gamma (supplied as a three-element vector argument to the SAT_P keyword), are required where:
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The map in the accompanying figure, which shows the eastern seaboard of the United States from an altitude of about 160km, above Newburgh, NY, was produced with the code that follows.
The parameters for this satellite projection are:
Labeling and drawing a vector on a satellite projection.
MAP_SET, /SATELLITE, SAT_P=[1.0251, 55, 150], 41.5, -74., $ /ISOTROPIC, /HORIZON, $ LIMIT=[39, -74, 33, -80, 40, -77, 41,-74], $ /CONTINENTS, TITLE='Satellite / Tilted Perspective' ; Set up the satellite projection: MAP_GRID, /LABEL, LATLAB=-75, LONLAB=39, LATDEL=1, LONDEL=1 ; Get North vector: p = convert_coord(-74.5, [40.2, 40.5], /TO_NORM) ; Draw North arrow: ARROW, p(0,0), p(1,0), p(0,1), p(1,1), /NORMAL XYOUTS, -74.5, 40.1, 'North', ALIGNMENT=0.5
