"The shortest distance between two points is a straight line" -- or is it? This familiar axiom rarely is true on maps of the world. The fact is, there are only a few ways of creating a world map so that the shortest distance between two points is shown by a straight line. Like so many other problems encountered in mapping, the difficulty stems from the simple fact that the earth is a curved surface and a map is flat: A map must depict a three-dimensional form in only two dimensions.
Imagine trying to flatten out a globe; you would have to stretch it here, compress it there, causing its scale to vary across the surface. Thus, it is quite common for directions and distances between pairs of locations to be represented unrealistically on maps. The amount of such distortion is significant only if the map shows a large portion of the earth's surface, such as a hemisphere or the entire world. At the scale of a city or even a small country, the inaccuracies caused by projecting the spherical surface to a flat page are not very great.
A great circle is a trace on the surface of the earth of a plane that passes through the center of the earth and divides it into halves. The most useful thing about a great-circle arc is that on the earth's surface, or on a map, it shows the shortest distance between points along that line. On some map projections, great-circle arcs are represented as straight lines, making them quite convenient to use for determining great-circle distances, directions, or courses. The following is a summary of some of those map projections and their individual characteristics.
The Azimuthal Equidistant Projection
The Azimuthal Equidistant projection may be centered on any point on the earth's surface. A straight line connecting that central point to any other point on the map will represent the shortest route between the two locations. Distances and directions along the line also will be correct; hence the term "equidistant" in the name of the projection.
If the central point is made the North or South Pole, then the parallels would be equally spaced concentric circles. It is usually more useful, however, to center the projection on a particular city or other point of interest, as has been done in Figure 9-1. A straight line from the map's central point (Miami, in this case) to Tokyo (or any other place) shows the shortest route and the correct distance and direction between the two. The central point could be any other significant place such as an airport, radio transmission facility, missile launch site, or seismographic station. The entire world can be shown on this projection, although shapes and sizes become very distorted near the outer edges.
The Gnomonic Projection
The Gnomonic projection (Figure 9-2) is another member of the azimuthal projection family (maps projected to a plane surface that is tangent to the globe at a single point), and it has the distinction of being the only map projection on which any straight line represents a great-circle arc. Whereas on the Azimuthal Equidistant projection only straight lines that originate at the map's central point show great-circle arcs, the Gnomonic projection has no such limitation. However, it does have a major restriction; it is not possible to show even a whole hemisphere on a Gnomonic projection.
Another problem is that shapes and sizes become greatly exaggerated within a relatively few degrees of the central point. Figure 9-2, for example, shows a Gnomonic projection centered on the North Pole and extending outward only 65° (to 25° N latitude), and it is evident how distorted the shapes have become.
With careful centering, however, the projection can be very useful for areas smaller than a hemisphere. In the example shown here, the great-circle path can be determined between any pair of points in North America, Europe, and much of Asia and North Africa.
These two projections, the Azimuthal Equidistant and the Gnomonic, probably are the most versatile and commonly used projections for representing great-circle routes. They are not the only alternatives, however.
Great Circles on Cylindrical Projections
Some projections show great-circle arcs as straight lines in more limited ways, such as in a single direction or along just one or two lines on the map. For example, the basic conic and cylindrical projections show north-south, great-circle courses (meridians) as straight lines, and on cylindrical projections the great circle formed by the equator also appears as a straight line.
It is possible to "force" two points to lie along one of the straight lines that is a great-circle arc on the projection. This necessitates "tilting the earth" so that the map is transverse (90°) or oblique (less than 90°) to the normal equatorial orientation of the projection. Figure 9-3, for example, shows a form of a cylindrical projection, the Plate Carrée. In this case the cylinder onto which the map was projected has been rotated around the globe at an oblique angle until Miami and Tokyo were aligned along the projection's "central meridian." On the normal aspect of the Plate Carrée, the great-circle arcs of the equator and all meridians are represented as straight lines that are true to scale.
As a further illustration of how different projection systems can affect the representation of shortest routes, look at Figure 9-4. This shows the normal aspect of another cylindrical projection, the Mercator, with the great-circle route between Miami and Tokyo plotted on it. The equator and the meridians are the only great-circle arcs this projection represents as straight lines.
Obviously, in its normal aspect, this is not a good projection for showing the shortest route between points on the earth's surface unless they happen to lie on the equator or a meridian. Otherwise, great-circle routes do not look like the most direct ones at all. To achieve a straight line for the great-circle arc between Miami and Tokyo with this projection, one could rotate the projection to a position where Miami and Tokyo lie on the "equator" of the projection, much as was done along the central meridian in Figure 9-3. Figure 9-5 shows the results of such an operation.
Representing great circles on maps is a special problem that can be solved only by using specialized projections. Formerly, the calculations and drafting skill required to produce projections manually undoubtedly deterred people from using a more appropriate projection for a given purpose. Now, fortunately, computer programs that generate a wide variety of different projections are available, and they make the task of transforming a map from one projection to another relatively painless. Nevertheless, the selection of an appropriate projection begins with a basic understanding of the map projections available!
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