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in mathematics. A mapping of a set A into a set B is a correspondence that associates to each element of x of the set A a definite element y = f(x) of B; y is called the image of x and x the preimage of y. Examples of mappings are parallel projection of a plane onto a plane and stereographic projection of a sphere onto a plane. A geographic map may be considered to be the result of mapping the points of the earth’s surface or part of it onto the points of a portion of a plane.
The concept of mapping is logically the same as the concepts of function, operator, and transformation. As a means of investigation, mapping makes it possible to replace the study of relations between the elements of set A by the study of relations between the elements of set B; in many cases, the latter may prove to be simpler. Thus, for example, any parallelogram can be mapped onto a square by a parallel projection and any quadratic curve can be mapped onto a circle by a central projection. Many properties are invariant, that is, are preserved, under mapping. For example, parallel projection preserves the parallelism of lines, ratios of lengths of segments of parallel lines, and other properties.
If every element of B is the image of an element of A, then the mapping is called a mapping of A onto B. If every element of B has one and only one preimage, then the mapping is said to be one-to-one. A mapping is called continuous if it carries close elements of A to close elements of B. More precisely, this means that if the elements x1, x2, …, xn, … converge to x, then the elements f(x1), f(x2), …, f(xn), … converge to f(x).
If T is a subset of A, then the subset f(T) of B that consists of the images of the points of T is called the image of T. If all points of a subset Q of B are the images of points in A, then the set of points x in A such that f(x) lies in Q is called the preimage of Q and is denoted by f−1(Q). In a one-to-one mapping, the preimage of each element of B is just one element of A.
A one-to-one mapping has an inverse mapping that associates to an element y in B its preimage f−1(y). A one-to-one mapping is said to be topological, or homeomorphic, if both it and its inverse mapping are continuous. Homeomorphisms (homeomorphic mappings) preserve only the most general properties of figures, such as connectivity, orientability, and dimensionality. Thus, a square and a circle are homeomorphic, but a square and a cube are not. Properties of figures that do not change under homeomorphisms are studied in topology. If certain relations hold in sets A and B and if these relations are preserved under a mapping, then the mapping is said to be an isomorphism with respect to these relations. (SeeISOMORPHISM.)
The mapping of a set of functions onto another plays a major role in mathematical analysis. For example, differentiation may be considered as a mapping that associates to a function f(x) the function f’(x). Among such mappings, the simplest are those that carry the sum of two functions to the sum of their images and the product of a function by a number to the product of its image by that number. Such mappings are said to be linear and are studied in functional analysis.
In many cases, it is possible to introduce coordinates in sets A and B; that is, each point in these sets can be given by a system of numbers (x1, …, xn) and (y1, …, ym). Then a mapping is given by a system of functions yk = fk(x1, …, xn), 1 ≤ k ≤ m. In most cases encountered in practice, the functions f1, f2, …, fm are differentiable; in such cases, the mapping is called differentiable. If the mapping is differentiable (m = n) and the Jacobian of the mapping is nonzero, then the mapping is one-to-one.
Differentiable mappings of surfaces onto surfaces are studied in differential geometry. There are properties common to all differentiable geometric mappings. For example, it is always possible to find an orthogonal net on the surface S whose image is an orthogonal net. on the surface S’. This theorem is of great importance in cartography.
The most important types of mappings of surfaces are isometric, conformal, spherical, geodetic, and area-preserving. An isometric mapping is characterized by the fact that any arc on S has the same length as its image on S’. Such mappings preserve the areas of figures and the angles between any two directions issuing from a point. A conformal mapping preserves the angles between any two directions issuing from a point. An example of such a mapping is a stereographic projection. A spherical, or Gauss, mapping of a surface S onto a sphere Σ associates to each point M of S a point M’ of Σ such that the normals to S’ and Σ at M and M’, respectively, are parallel. More generally, we can consider a mapping of S onto a surface S’ such that the normals at corresponding points are parallel. In a geodetic mapping of surfaces S and S’, the image of a geodesic on 5 is a geodesic on S’. Geodetic mapping of a surface of constant negative curvature onto part of a plane is of great importance for interpreting Lobachevskian geometry. In an area-preserving, or equiareal, mapping of a surface onto a surface, the areas of corresponding figures are equal.
From the standpoint of cartography, each of the three mappings of a curved surface onto a plane—conformal, geodetic, and area-preserving—has advantages. It can be shown that it is impossible to satisfy all these requirements or even any two at the same time.
REFERENCESRashevskii, P. K. Rimanova geometriia i tenzornyi analiz, 3rd ed. Moscow, 1967.
Blaschke, W. Differentsial’naia geometriia igeometricheskie osnovy teorii otnositel’nosti Einshteina, part 1. Moscow-Leningrad, 1935. (Translated from German.)
Hilbert, D., and S. Cohn-Vossen. Nagliadnaia geometriia, 2nd ed. Moscow-Leningrad, 1951. (Translated from German.)
mapping(1) See map, mapping app and digital mapping.
(2) Translating or converting one set of values to another set. See map.
(3) Identifying one set of values with another set. See map.
(4) Assigning hardware, software or attributes to users or user groups. See map.