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In the 1942 issue of the Geographical Review published by the American Geographical Society, Osborne Maitland Miller presented a map projection which might be used as an alternative to that devised by Mercator. This is now known as Miller's Cylindrical projection and is quite popular. Its advantages are that it provides a good approximation to the Mercator near the Equator and has a finite limit at the poles. Of course, it is not conformal, nor are loxodromes represented by straight lines. Miller actually investigated several projections, comparing seven of them to Mercator's, and chose the one described here as the best for his purpose. This purpose, proposed by S. Boggs, the then Geographer to the U. S. Department of State, was to find an acceptable balance between shape and area distortion. I present here two much simpler alternatives which have properties similar to Miller's projection. As with all cylindrical projections, the equations are of the form: (1) x = [Lambda] (2) y = f([Psi]) where [Psi] and [Lambda]. are the latitude and longitude measured in radians. The latitudinal functions for Mercator's, Miller's, and the two new variant projections are as follows: (3) Mecator= Ln Tan([Pi]/4 + [Psi]/2) (4) Miller 1.25 Ln Tan ([Pi]/4 + 2 [Psi]/5) (5) Variation # 1 = [Psi] + 1/6[[Psi].sup.3] (6) Variant # 2 = [Psi] + 1/6[[Psi].sup.3] + 1/24 [[Psi].sup.5] These projections agree almost completely to latitude 45 degrees. The graph in Figure 1 compares the latitudinal spacing beyond 60 degrees, where the difference becomes noticeable. As can be seen on the graph, the first variant is virtually identical to Miller's projection, and the second is closer to Mercator's projection than either the Miller's or the variant 1 projections are. [Figure 1 ILLUSTRATION OMITTED] The equations for the two variant projections are much simpler than those for Miller's (and for Mercator's) projection, have simpler inverses, and simpler derivatives for the calculation of distortion. These advantages are of benefit primarily for computational purposes, as the differences are hardly detectable on a world map without actual measurement (see Figure 2). It is of course ironic that Mercator's anamorphose, designed for the graphical solution to a problem in analytical geometry on a sphere, should even be considered for a non-navigational world map. None of the alternatives to Mercator's projection are suitable for navigation. [Figure 2 ILLUSTRATION OMITTED] We should ask how well any of these projections satisfy Boggs' criteria. To quote from Miller's article (1942, p. 425): By an "acceptable" balance is meant one which to the uncritical eye does not obviously depart from the familiar shapes of the land areas as depicted by the Mercator projection but which reduces areal distortion as far as possible under these conditions. It is rather difficult to convert this curious statement into a well defined system of equations to be solved. What, for example, is an `uncritical eye'? Does this mean we can tolerate an error of 5%, or 10%, or what? Reducing areal distortion "as far as possible" can take several meanings. George Airy's (1861) criterion of balancing the world average squared areal and angular distortions leads to the well defined, though not simple, problem of finding a function which minimizes an integral that is a linear combination of the two types of distortion, subject to appropriate constraints. This is a problem in the calculation of variations that has led to the so called "optimal" projections (James and Clarke 1862, Young 1920, Mescheryakov 1965, Biernacki 1965, Milnor 1969, Baetsle 1970, Gilbert 1974, Tobler 1977, Peters 1981, Pearson 1982, Hoschek 1984, Canters 1989, Laskowski 1991). Two recent studies (Grafarend and Niermann 1984, Gyorffy 1990) have applied Airy's criteria to cylindrical projections. … |
