Using IDL: Map Projections

## Cylindrical Projections

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 Projection

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. Therefore, uses include navigation maps and equatorial maps. 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. Limits are typically specified because of the great distortions around the poles when the equator is selected.

A simple mercator projection with latitude ranges from -80 degrees to 80 degrees is shown in the following figure.

Figure 9-9: Simple Mercator Projection

### Transverse Mercator Projection

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.

The following Transverse Mercator map shows North and South America, with a central meridian of -90 degrees West and centered on the Equator.

Figure 9-10: Transverse Mercator Projection

### Cylindrical Equidistant Projection

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

 -180 £ u £ 180 and -90 £ v £ 90

If EQ is the equator, meridians and parallels will be equally spaced parallel lines.

The following figure shows a simple cylindrical equidistant projection and an oblique cylindrical equidistant projection rotated by 45°.

Figure 9-11: Cylindrical Projections

### Miller Cylindrical Projection

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.

### Conic Projection

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 can be specified and are projected onto circles and meridians onto equally spaced straight lines. The following figure shows the map shown in the accompanying figure, which features North America with standard parallels at 20 degrees and 60 degrees.

Figure 9-12: Lambert's Conformal Conic with Standard Parallels at 20° and 60°

### Albers Equal-Area Conic Projection

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.