Mathematical Cartography

a cartographic discipline that studies the theory of map projections and of transformations of map projections, methods of obtaining projections, and ways of making efficient use of various projections in practice. Sometimes the term “mathematical cartography” is used to denote all the problems arising in the mathematics of map-making (the map layout, margins), as well as ways and means of carrying out measurements on maps. Mathematical cartography is closely related to mathematics, geodesy, cartography, and other disciplines.

During the first stages of the development of mathematical cartography (sixth century B.C. to the 17th century), various map projections were invented, investigated, and used. From the 18th century to the early 20th, separate classes of projections were studied. The theory of creating new methods of obtaining different (in part, new) classes or groups of projections and the theory of their transformations have been developing since the mid-20th century. The methods of modern mathematical cartography have been mechanized and automated, and computers are sometimes used.

Direct and inverse problems are distinguished in mathematical cartography. The direct problem of mathematical cartography is the study of the properties of map projections defined by equations of the form

(1) x = f₁ (φ, λ) y = f₂ (φ, λ)

where φ and λ are the latitude and longitude, respectively, of a point on an ellipsoid representing the earth. This problem is solved using formulas from the theory of distortions.

The inverse problem of mathematical cartography consists in obtaining the equations (1) or, more generally, in finding the projections corresponding to given distortion distributions.

Different methods of constructing projections have been used throughout the history of mathematical cartography: geometrical, analytic, graphic-analytic, and other methods. However, these methods yielded particular projections and rather special sets of projections. A general method of finding projections, which at the same time yields a solution to the inverse problem of mathematical cartography, is derived from the Euler-Urmaev system

where m and n are the meridian and parallel scales and ε the angle between their images; β is the meridian convergence. This set of equations is a system of two first-order quasilinear partial differential equations (for example, nφ = ∂n/∂φ). It is subdefinite since there are two equations and four functions.

Different methods of completing the definition of the system (2), based on an a priori prescription of a distortion distribution needed in practice, make it possible to study every possible class of projections. From the standpoint of analysis, the system (2) yields necessary and sufficient existence conditions for projections with specified distortion distributions. The system (2) and the formulas in distortion theory, and certain modifications of them, are referred to as the fundamental equations of mathematical cartography. Methods of numerical analysis, the theory of conformal and quasiconformal mappings, the calculus of variations, and other branches of mathematics are widely used in finding new projections.

The system (2) leads to the most complete genetic classification of map projections that includes all known and conceivable projections. It is based on the concept of a class of projections as a set of projections such that [after completing the definition of system (2) by equations of the projections in the characteristics] the set is described by a definite system of two first-order partial differential equations. Relevant examples are the class of conformal projections and the class of Euler projections. Systems of classes of projections may be elliptical, hyperbolic, or of some other type, and the projections described by them are of the corresponding type. This fact is of fundamental importance when we are looking for projections in a particular class, for it enables us to predict certain properties of new projections. Thus, mathematical cartography is a unique “storehouse” of the science of cartography and actual map-making, whose “compartments” contain definite classes and other sets of map projections. The required projection can be taken from it (or a new one found) for a concrete productive task.

One of the central problems in mathematical cartography is the task of constructing the best map projections, that is, projections in which the distortions are reduced in some sense to a minimum. It still has not been entirely solved even for very well-known classes of projections, although many famous scientists (L. Euler, K. Gauss, P. L. Chebyshev) have studied particular cases of this problem. The problem is posed in two ways: for a given region it is necessary to find projections with a minimum of distortion either from the entire conceivable set of projections (ideal projections) or from a specified class (best projections of a class). In both cases the problem from a mathematical standpoint becomes that of approximating functions of two variables. However, for the second case there also exist different formulations. For example, in proceeding on the basis of the theory of best approximations, we speak of best minimax projections, and in using the theory of mean square approximations we investigate the best variational projections.

The general problem of constructing the best map projections reduces to a number of new extremum problems for a conditional minimax. Only the case of best conformal projections has been investigated thoroughly. According to the ChebyshevGrave theorem, the best conformal projection (Chebyshev) for a given region is that for which the extreme isocolon at which a given projection coincides with the contour of the area to be mapped. In Chebyshev projections distortions of areas deviate least from zero. As a corollary, the moduli of the logarithms of the length scales also deviate least from zero. The ratio of the largest scale to the smallest is minimal. The largest curvature of the images of the geodesies is also minimal. Finally, the standard deviation of the logarithms of the lengths scale is minimal. Such a combination of favorable properties in Chebyshev projections is typical for the class of conformal projections, which is the simplest though most important projection in practice. An example of a Chebyshev projection is a stereographic projection, which satisfies the conditions of the theorem when we map a spherical segment onto a plane and choose a suitable value of a certain arbitrary constant. The technique of constructing Chebyshev projections has been developed in detail for arbitrary areas. The Chebyshev-Grave theorem is valid for a number of other particular classes of projections that are not conformal but of elliptical type.