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Charts and Projections (Part Two)

in Maps and Charts

Gnomonic Projection


All gnomonic projections are direct perspective projections. Since the plane of every great circle cuts through the center of the sphere, the point of projection is in the plane of every great circle. This property then becomes the most important and useful characteristic of the gnomonic projection. Each and every great circle is represented by a straight line on the projection. A complete hemisphere cannot be projected onto this plane because points 90° from the center of the map project lines parallel to the plane of projection. Because the gnomonic is nonconformal, shapes or land masses are distorted, and measured angles are not true. At only one point, the center of the projection, are the azimuths of lines true. At this point, the projection is said to be azimuthal. Gnomonic projections are classified according to the point of tangency of the plane of projection. A gnomonic projection is polar gnomonic when the point of tangency is one of the poles, equatorial gnomonic when the point of tangency is at the equator and any selected meridian. [Figure 1-21]

Figure 1-21. Polar gnomonic and stereographic projections.

Figure 1-21. Polar gnomonic and stereographic projections. [click image to enlarge]

Stereographic Projection

The stereographic projection is a perspective conformal projection of the sphere. The term oblique stereographic is applied to any stereographic projection where the center of the projection is positioned at any point other than the geographic poles or the equator. If the center is coincident with one of the poles of the reference surface, the projection is called polar stereographic. The illustration in Figure 1-21 shows both gnomonic and stereographic projections. If the center lies on the equator, the primitive circle is a meridian, which gives the name meridian stereographic or equatorial stereographic.

Cylindrical Projections

The only cylindrical projection used for navigation is the Mercator, named after its originator, Gerhard Mercator (Kramer), who first devised this type of chart in the year 1569. The Mercator is the only projection ever constructed that is conformal and, at the same time, displays the rhumb line as a straight line. It is used for navigation, for nearly all atlases (a word coined by Mercator), and for many wall maps.

Imagine a cylinder tangent to the equator, with the source of projection at the center of the earth. It would appear much like the illustration in Figure 1-22, with the meridians being straight lines and the parallels being unequally spaced circles around the cylinder. It is obvious from Figure 1-22 that those parts of the terrestrial surface close to the poles could not be projected unless the cylinder was tremendously long, and the poles could not be projected at all.

Figure 1-22. Cylindrical projection.

Figure 1-22. Cylindrical projection.

On the earth, the parallels of latitude are perpendicular to the meridians, forming circles of progressively smaller diameters as the latitude increases. On the cylinder, the parallels of latitude are shown perpendicular to the projected meridians but, since the diameter of a cylinder is the same at any point along the longitudinal axis, the projected parallels are all the same length. If the cylinder is cut along a vertical line (a meridian) and spread flat, the meridians appear as equally spaced, vertical lines, and the parallels as horizontal lines, with distance between the horizontal lines increasing with distance away from the false (arbitrary) meridian.

The cylinder may be tangent at some great circle other than the equator, forming other types of cylindrical projections. If the cylinder is tangent at a meridian, it is a transverse cylindrical projection; if it is tangent at any point other than the equator or a meridian, it is called an oblique cylindrical projection. The patterns of latitude and longitude appear quite different on these projections because the line of tangency and the equator no longer coincide.

Mercator Projection

The Mercator projection is a conformal, nonperspective projection; it is constructed by means of a mathematical transformation and cannot be obtained directly by graphical means. The distinguishing feature of the Mercator projection among cylindrical projections is: At any latitude the ratio of expansion of both meridians and parallels is the same, thus, preserving the relationship existing on the earth. This expansion is equal to the secant of the latitude, with a small correction for the ellipticity of the earth. Since expansion is the same in all directions and since all directions and all angles are correctly represented, the projection is conformal.

Figure 1-23. Mercator is conformal but not equal area.

Figure 1-23. Mercator is conformal but not equal area.

Rhumb lines appear as straight lines and their directions can be measured directly on the chart. Distance can also be measured directly, but not by a single distance scale on the entire chart, unless the spread of latitude is small. Great circles appear as curved lines, concave to the equator or convex to the nearest pole. The shapes of small areas are very nearly correct, but are of increased size unless they are near the equator. [Figure 1-23] The Mercator projection has the following disadvantages:

  1. Measuring large distances accurately is difficult.
  2. Must apply conversion angle to great circle bearing before plotting.
  3. Is useless above 80° N or below 80° S since the poles cannot be shown.

The transverse or inverse Mercator is a conformal map designed for areas not covered by the equatorial Mercator. With the transverse Mercator, the property of straight meridians and parallels is lost, and the rhumb line is no longer represented by a straight line. The parallels and meridians become complex curves and, with geographic reference, the transverse Mercator is difficult to use as a plotting chart. The transverse Mercator, though often considered analogous to a projection onto a cylinder, is in reality a nonperspective projection, constructed mathematically. This analogy, however, does permit the reader to visualize that the transverse Mercator shows scale correctly along the central meridian, which forms the great circle of tangency. [Figure 1-24] In effect, the cylinder has been turned 90° from its position for the ordinary Mercator, and a meridian, called the central meridian, becomes the tangential great circle. One series of NGA charts using this type of projection places the cylinder tangent to the 90° E–90° W longitude.

Figure 1-24. Transverse cylindrical projection—cylinder tangent at the poles.

Figure 1-24. Transverse cylindrical projection—cylinder tangent at the poles. [click image to enlarge]

These projections use a fictitious graticule similar to, but offset from, the familiar network of meridians and parallels. The tangent great circle is the fictitious equator. Ninety degrees from it are two fictitious poles. A group of great circles through these poles and perpendicular to the tangent constitutes the fictitious meridians, while a series of lines parallel to the plane of the tangent great circle forms the fictitious parallels.

On these projections, the fictitious graticule appears as the geographical one ordinarily appearing on the equatorial Mercator. That is, the fictitious meridians and parallels are straight lines perpendicular to each other. The actual meridians and parallels appear as curved lines, except the line of tangency. Geographical coordinates are usually expressed in terms of the conventional graticule. A straight line on the transverse Mercator projection makes the same angle with all fictitious meridians, but not with the terrestrial meridians. It is, therefore, a fictitious rhumb line.

The appearance of a transverse Mercator using the 90° E–90° W meridian as a reference or fictitious equator is shown in Figure 1-24. The dotted lines are the lines of the fictitious projection. The N–S meridian through the center is the fictitious equator, and all other original meridians are now curves concave on the N–S meridian with the original parallels now being curves concave to the nearer pole. To straighten the meridians, use the graph in Figure 1-25 to extract a correction factor that mathematically straightens the longitudes.

Figure 1-25. Transverse Mercator convergence graph.

Figure 1-25. Transverse Mercator convergence graph. [click image to enlarge]

 

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