MAP TRANSFORMATIONS OF GEOGRAPHIC SPACE by WALDO

Many geographic and economic models of human behavior in a spatial context indicate that the measuring rod of the geodesist or surveyor is less relevant than a scaling of distance in temporal or monetary units. It is necessary to take into account not only the shape of the earth but also the realities of transportation on this surface. Automobiles, trains, airplanes, and other media of transport can be considered to have the effect of modifying distances-measured in temporal or monetary units-in a complicated manner. Different distance relations, however, can be interpreted as different types of geometry. A geographically natural approach is to attempt to map this geometry to a plane, in a manner similar to the preparation of maps of the terrestrial surface. If the transformations were dependent only on a factor of proportionality, as in the conversion from miles to kilometers, there would be no difficulty. The facts are considerably more complex. The geometry with which we must deal is rarely Euclidean, and it is, in general, impossible to prepare a plane map preserving all distance relations, just as isometric transformations of a spherical surface to a plane are not possible. Maps preserving distance relations from one point are easily achieved, however, whatever the units of measurement. The transformations are approached from the theory of map projections, as available and familiar to geographers. Numerous transformations preserving time or cost distance relations from (or to) one location as found in actual transport systems or situations can be achieved as azimuthal projections by relaxing the traditional assumptions applied to this class of projections. The parametric classification of map projections employed suggests a simple graphic technique for transformations of empirical data recorded in the form of isochrones or isovectures. Though yielding a simple method of obtaining polar isometric transformations, azimuthal projections are less appropriate than polar geodesic isometries. Neither polar isometric nor polar geodesic transformations are unique, and it is natural to inquire whether further conditions may be applied. Consideration is given to equal-area maps, conformal maps, and measures of the departure of an environment from an idealized isotropic geographic space, the latter from Tissot's work. Complicating factors are that time or cost distances in general do not satisfy the postulates of a metric space, they vary frequently, and are commodity specific. The differential resource distribution on the surface of the earth also suggests that equal-area maps are not always the most …