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A cylindrical projection maps the globe to a cylinder which is formed by wrapping the UV plane around the globe with the u-axis coinciding with a great circle. The parameters P0lat, P0lon, and Rot determine the great circle that passes through the point C=(P0lat, P0lon). In the discussions below, this great circle is sometimes referred to as EQ. Rot is the angle between North at the map's center and the v-axis (which is perpendicular to the great circle). The cylinder is cut along the line parallel to the v-axis and passing through the point diametrically opposite to C. It is then rolled out to form a plane.
The cylindrical projections in IDL include: Mercator, Transverse Mercator, cylindrical equidistant, Miller, Lambert's conformal conic, and Alber's equal-area conic.
Mercator's projection is partially developed by projecting the globe onto the cylinder from the center of the globe. This is a partial explanation of the projection because vertical distances are subjected to additional transformations to achieve conformity-that is, local preservation of shape. To properly use the projection, the user should be aware that the two points on the globe 90 degrees from the central great circle (e.g., the North and South Poles in the case that the selected great circle is the equator) are mapped to infinite distances. By default, the keyword LIMIT is set to [-80, -180, 80, 180] because of the great distortions around the poles when the equator is selected.
The following statement produces a simple Mercator projection:
MAP_SET, /MERCATOR, 0, 0, /ISOTROPIC, $ /GRID, /CONTINENTS, $ TITLE = 'Simple Mercator'
The result of this statement is shown in the upper-left corner of the following figure.
Latitudes range from -80 degrees to 80 degrees.
The Transverse Mercator (also called the UTM, and Gauss-Krueger in Europe) projection rotates the equator of the Mercator projection 90 degrees so that it follows a specified central meridian. In other words, the Transverse Mercator involves projecting the Earth onto a cylinder which is always in contact with a meridian instead of with the Equator.
The central meridian intersects two meridians and the Equator at right angles; these four lines are straight. All other meridians and parallels are complex curves which are concave toward the central meridian. Shape is true only within small areas and the areas increase in size as they move away from the central meridian. Most other IDL projections are scaled in the range of +/- 1 to +/- 2 Pi; the UV plane of the Transverse Mercator projection is scaled in meters. The conformal nature of this projection and its use of the meridian makes it useful for north-south regions.
The Clarke 1866 ellipsoid is used for the default, but its parameters can be altered with the ELLIPSOID keyword.
To create a UTM map, centered near London, with a scale of 10 million to one, type the following:
MAP_SET, /TRANSVERSE, 51, 0, SCALE=10e6, $
/GRID, LATDEL=2.5, LONDEL=2.5, /LABEL, LONLAB=48, $
/CONTINENTS, E_CONT={COUNTRIES:1, COASTS:1}, $
TITLE='UTM Projection'
When the eccentricity of the Earth is not important, global scale Transverse Mercator projections can be easily created using the Mercator projection with the CENTRAL_AZIMUTH keyword set to 90 degrees, and setting Rot to rotate the map 90 degrees. For example, to create the Transverse Mercator map showing North and South America, with a central meridian of -90 degrees West and centered on the Equator, shown in the upper-right corner of the figure in the Mercator Projection section. It is produced by the following statement:
MAP_SET, /MERCATOR, 0, -75, 90, CENTRAL_AZIMUTH=90, $ /ISOTROPIC, LIMIT= [32,-130, 70,-86, -5,-34, -58, -67], $ /GRID, LATDEL=15, LONDEL=15, /CONTINENTS, $ TITLE = 'Transverse Mercator'
The cylindrical equidistant projection is one of the simplest projections to construct. If EQ is the equator, this projection simply lays out horizontal and vertical distances on the cylinder to coincide numerically with their measurements in latitudes and longitudes on the sphere. Hence, the equidistant cylindrical projection maps the entire globe to a rectangular region bounded by
If EQ is the equator, meridians and parallels will be equally spaced parallel lines.
The following code is used to produce a simple cylindrical equidistant projection and an oblique cylindrical equidistant projection as shown in the lower-left and lower-right sections of the figure under the Mercator Projection heading:
MAP_SET, /CYLINDRICAL, 0, 0, /GRID, /CONTINENTS, $ TITLE = 'Simple Cylindrical Equidistant'
Now rotate the projection by 45%:
MAP_SET, /CYLINDRICAL, 0, 0, 45, $ /GRID, /CONTINENT, /HORIZON, $ TITLE='Oblique Cylindrical Equidistant'
The Miller projection is a simple mathematical modification of the Mercator projection, incorporating some aspects of cylindrical projections. It is not equal-area, conformal or equidistant along the meridians. Meridians are equidistant from each other, but latitude parallels are spaced farther apart as they move away from the Equator, thereby keeping shape and area distortion to a minimum. The meridians and parallels intersect each other at right angles, with the poles shown as straight lines. The Equator is the only line shown true to scale and free of distortion.
The Lambert's conformal conic with two standard parallels is constructed by projecting the globe onto a cone passing through two parallels. Additional scaling achieves conformity. The pole under the cone's apex is transformed to a point, and the other pole is mapped to infinity. The scale is correct along the two standard parallels. Parallels are projected onto circles and meridians onto equally spaced straight lines. The STANDARD_PARALLELS keyword specifies the latitudes of one or two standard parallels.
The following statement produces the map shown in the accompanying figure, which features North America with standard parallels at 20 degrees and 60 degrees:
MAP_SET, /CONIC, 40, -80, STANDARD_PARALLELS=[20,60], $ /ISOTROPIC, LIMIT=[0, -260, 90, 100], $ /GRID, LATDEL=15, LONDEL=20, /CONTINENT, $ TITLE= 'Lambert's Conic'
The Albers Equal-Area Conic is like most other conics in that meridians are equally spaced radii, parallels are concentric arcs of circles and scale is constant along any parallel. To maintain equal area, the scale factor along meridians is the reciprocal of the scale factor along parallels, with the scale along the parallels between the two standard parallels too small, and the scale beyond the standard parallels too large. Standard parallels are correct in scale along the parallel, as well as in every direction.
The Albers projection is particularly useful for predominantly east-west regions. Any keywords for the Lambert conformal conic also apply to the Albers conic.
