For some maps, we may be less concerned about the great-circle path between two points than we are about the direction and distance from one point to another. For radio signals, for example, these values affect the pointing of a directional transmitter or receiver as well as the strength of the signal.
Although in general it is not practical to measure distance and direction accurately between any random pair of points on a map, there are some projections that are very useful for such measurements when one of the points is always the same. The most familiar such projection is the oblique Azimuthal Equidistant, centered at any desired point on the earth, such as Washington, D.C. (Figure 11-1), and discussed in Chapter 9. In this case all distances can be measured from Washington to any other point by connecting the two points with a straightedge and applying the map scale to the measured distance. The direction from Washington is obtained by measuring the angle that this straight line (which is also the great-circle path) makes at Washington with the direction to the North Pole, along the central meridian.
The Oblique Werner Projection
The Azimuthal Equidistant projection does not maintain true areas, but another projection has this property and also shows the correct distance. This is an oblique form (Figure 11-2) of the Werner projection, a heart-shaped projection originally devised and used during the 16th century, but rarely used since then. In this case, however, the straight line connecting the center and some other point, although having the correct length, does not follow the great-circle route, and its direction cannot be determined readily.
Another way to measure distance and direction is to treat them both as rectangular coordinates by plotting an oblique aspect of another very old projection called the Plate Carrée or Plane Chart (Figure 11-3), discussed before (see chapters 9 and 10). In the normal case all meridians and parallels are equally spaced straight lines, and the scale is correct along all meridians and the equator. If the point of interest, such as Washington, is placed at the "north pole" of this projection (Figure 11-4), that point becomes the entire top edge of the map, and its antipode, the bottom edge. Then the great-circle route from Washington to any second point is the straight vertical line intersecting the second point, while the distance is that directly measured from the top edge. The direction from Washington to the second point is directly proportional to the horizontal distance between that vertical line and the vertical line on which the North Pole lies, the total horizontal extent being 360°.
It is also possible to have a projection on which a straight line connecting the center and any other point shows the correct distance, and on which the direction from the second point to the center relative to north is the same as the angle between the connecting line and a line extending straight up from the second point. The direction from the center to the second point, however, will generally not be correct. This is called a retroazimuthal projection, and this form was devised by Hammer in Germany in 1910. In 1929, it was extended in England to a world map, inspired by the desire for a map to allow persons throughout the world to determine the directions to point antennas on radio receivers in order to pick up transmission from a powerful station in Rugby, England. The far side of the earth must be plotted backwards, and the entire projection has a strange shape. Figure 11-5 shows the two parts of a world map centered near St. Louis, Mo. They should be superimposed.
The first retroazimuthal projection was devised by Craig in Egypt in 1909 as the Mecca projection, so-called because with it Moslems anywhere could determine which way to face toward Mecca (the center) for prayers. Craig's retroazimuthal projection is different from Hammer's in that meridians are straight and vertical instead of curved, and distance cannot be measured readily, but direction is measured as on Hammer's.
Measuring from Two Points
There is one projection on which all directions (but not distances) are correct from not just one but two points. It is logically called the Two-Point Azimuthal projection and was first presented by Maurer of Germany in 1914. It is the same as an oblique Gnomonic projection that has been compressed uniformly in one direction. Since all great circles are straight lines on the original Gnomonic, they are all straight on the Two-Point Azimuthal projection. On a related but different projection, presented by Maurer in 1919, the Two-Point Equidistant (Figure 11-6), all distances (but not directions) are correct from two points. On this projection, as on the oblique Werner above, the straight connecting lines do not represent the shortest routes from the two central points.
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