ThesisPDF Available

Die vorteilhaftesten Abbildungen in der Atlaskartographie

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Abstract

In der Thesis wurden erstmals die für Erdkarten häufig angewandten Abbildungen nach der mittleren quadratischen Längenverzerrungen verglichen. Da in den Abbildungen, in denen der Pol zu einer Linie entartet, Längenverzerrungen länges der Pollinie unendlich sind, können die Pole nicht in die Berechnungen einbezogen werden. Die Untersuchungen werden deshalb auf das Gebiet zwischen den Breitenkreisen φ = ±85º begrenzt. Die mittleren quadratischen Längenverzerrungen im gesamten Abbildungsgebiet werden nach den Kriterien von Airy und Airy-Kavrajski für alle zur Untersuchung ausgewählten Abbildungen berechnet. Durch das Umbeziffern von Kartennetzen werden weiter nach den beiden Kriterien die besten Abbildungen aus den verschiedenen für Erdkarten geeigneten Abbildungsgruppen entwickelt. Um die praktische Brauchbarkeit diser neuen Varianten zu prüfen und sie mit schon bekannten Abbildungen auch nach der Grösse und Verteilung von Verzerrungen vergleichen zu können, werden für alle diese Varianten die Abbildungsmaβstäbe in Richtung der Meridiane und Breitenkreise berechent und die Äquideformaten-Modelle der Flachenmaβstäbe p und Maximalwinkelverzerrungen ω erstelt. Alle Äquideformaten-Modelle (kartographische Netze mit Konturen der Kontinente und Äquideformaten p und ω) werden auf dem Plotter gezeichnet.
... Since here the spherical Earth will be mapped, the surface integral mentioned above turns into a double integral of the function 1 2 K · cos w. Furthermore, to avoid the distortion values tending to infinity at and near the poles, the values of the function above assigned to the points of the 5 • − 5 • environment of poles will be omitted from the double integral (Frančula, 1971;Gede, 2011, p. 218;Grafarend & Niermann, 1984, p. 104). So the final formula for E 2 K is: ...
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Chapter
Harmonic maps are a certain kind of an optimal map projection which has been developed for map projections of the sphere. Here we generalize it to the “ellipsoid of revolution”. The subject of an optimization of a map projection is not new for a cartographer. For instance, in Sect. 5-25, we compute the minimum distortion energy for mapping the “sphere-to-plane”.
Chapter
Conventionally, conformal coordinates, also called conformal charts, representing the surface of the Earth or any other Planet as an ellipsoid-of-revolution, also called the Geodetic Reference Figure, are generated by a two-step procedure. First, conformal coordinates (isometric coordinates, isothermal coordinates) of type UMP (Universal Mercator Projection, compare with Example 15.1) or of type UPS (Universal Polar Stereographic Projection, compare with Example 15.2) are derived from geodetic coordinates such as surface normal ellipsoidal longitude/ellipsoidal latitude. UMP is classified as a conformal mapping on a circular cylinder, while UPS refers to a conformal mapping onto a polar tangential plane with respect to an ellipsoid-of-revolution, an azimuthal mapping.
Chapter
In this chapter, we present a collection of most widely used map projections in the polar aspect in which meridians are shown as a set of equidistant parallel straight lines and parallel circles (parallels) by a system of parallel straight lines orthogonally crossing the images of the meridians. As a specialty, the poles are not displayed as points but straight lines as long as the equator. First, we derive the general mapping equations for both cases of (i) a tangent cylinder and (ii) a secant cylinder and describe the construction principle.
Chapter
At the beginning of this chapter, let us briefly refer to Chap. 8, where the data of the best fitting “ellipsoid-of-revolution to Earth” are derived in form of a table. Here, we specialize on the mapping equations and the distortion measures for mapping an ellipsoid-of-revolution \(\mathbb{E}_{A_{1},A_{2}}^{2}\) to a cylinder, equidistant on the equator. Section 14-1 concentrates on the structure of the mapping equations, while Sect. 14-2 gives special cylindric mappings of the ellipsoid-of-revolution, equidistant on the equator. At the end, we shortly review in Sect. 14-3 the general mapping equations of a rotationally symmetric figure different from an ellipsoid-of-revolution, namely the torus.
Chapter
A special mapping, which was invented by Gauss (1822, 1844), is the double projection of the ellipsoid-of-revolution to the sphere and from the sphere to the plane. These are conformal mappings. A very efficient compiler version of the Gauss double projection was presented by Rosenmund (1903) (ROM mapping equations) and applied for mapping Switzerland and the Netherlands, for example. An alternative mapping, called “authalic”, is equal area, first ellipsoid-of-revolution to sphere, and second sphere to plane.
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