Card design theory

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The map design theory (also map network design theory ) comprises those mathematical methods that were developed for the design of exact map networks and for the calculation of geodetic maps. It is a branch of mathematical cartography , but is also assigned to theoretical geodesy .

A map network design consists of a clear regulation as to how the curved surface of the earth with its latitude and longitude circles is to be mapped onto the flat surface of a map or plan. One achieves this

The simplest graphic map network: the square or rectangular flat map from the 1st century. The red lines show the true distances of the meridians.
  • graphically through a geometric construction
  • or arithmetically (analytically) through a system of mathematical formulas.

The result is a network of crossing lines that correspond to the circles of latitude and longitude on Earth. This inevitably leads to distortions because a double-curved surface (the globe or the earth's ellipsoid ) cannot be transferred to a plane without changing its shape - see the tearing of an orange peel when it is pressed flat on the plate.

The first graphic map network designs have been handed down from ancient Greece : the square flat map and the stereographic projection . At the turn of the century, simple computational methods were already being developed. If the map network is now available, any number of points or lines on the earth's surface can be transferred into the network through their geographical latitude and longitude .

The simplest map network is a rectangular grid of widths and lengths, the so-called square flat map . But if you extend the grid from the equator up to a width of 90 °, the poles become lines that are as long as the equator. To avoid such distortions, Gerhard Mercator devised a right-angled map network around 1600, but whose latitudes are increasingly spaced from the poles. This Mercator projection enlarges the surfaces, but ensures their correct shape ( angular accuracy ).

The strictly mathematical theory of map design goes back to Nicolas Auguste Tissot , who developed the theory of map distortion around 1850. The Tissot indicatrix indicates the ellipse to which a small circle on the globe deforms when it is projected onto the map with the selected formulas. Tissot's theory can also be used to determine those formulas that produce a desired property of the map projection. In this way z. B. calculates how the earth (the "archetype") is to be mapped onto a cone if a flight route across the Atlantic should appear distorted as little as possible. Similar methods are used in geodesy in order to be able to convert the survey points into digital coordinates as well as possible . A widely used method for this is the Gauß-Krüger projection . It changes the distances between the points slightly, but leaves the angles unchanged (see Conformity ).

In terms of geography, on the other hand, it is important to reproduce the areas of the countries as precisely as possible. The theory of map design knows a number of exactly equal-area projections, from which one can choose the most suitable. It will look different for a long country like Chile than for a very large country or even an entire hemisphere . However, angular and area fidelity are mutually exclusive, so that certain changes in the shape of borders or coasts have to be accepted in geography . Also, length accuracy can only be achieved in certain directions, and never in north-south and east-west at the same time.