From North to South.
This sequence of images starts with the equidistant azimuthal projection onto the tangent plane at the north pole. This is gradually changed to equidistant conical projections with the standard parallel moving towards the equator. So we arrive at the equidistant cylindrical projection.
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As the standard parallel moves further to the south, we finally obtain the equidistant azimuthal projection onto the tangent plane in the south pole. Tissot's Indicatrix.
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An easy way of visualising the distortion of a map at a given point P is Tissot's indicatrix , which is an ellipse centred at P. If the indicatrix is a circle then the map is conformal angle preserving at P, otherwise the main axes of the ellipse give the directions of maximal and minimal distortion.
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In our example you can see indicatrices for Sanson's projection. It is worth noting that all indicatrices, though they of different shape, have the same area. This illustrates the fact that Sanson's projection is area preserving. In mathematics , the words mapping , map , and transformation tend to be used interchangeably.
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The mathematical notion of mapping is an abstraction of the process of making a geographical map. It is now considered to be a fundamental notion pervading much of mathematics. Important special classes of mappings are homomorphisms in algebra , isometries in geometry , operators in analysis , homeomorphisms in topology , representations in group theory , and isomorphisms in a variety of contexts see foundations of mathematics: Isomorphic structures.
You are using an outdated browser. Please upgrade your browser to improve your experience and security. Info Print Cite. The direct problem of mathematical cartography is the study of the properties of map projections defined by equations of the form.
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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.
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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. 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.
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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.
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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. 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.