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.
... If one would like to reduce both areal and angular distortions at the same time, the equidistant cylindrical mapping would be the best choice (Györffy, 1990). All three variants may have two distortion-free parallels symmetrical to the Equator, the optimal choice (resulting in the least distortion possible) was observed to be the same for all three variants (Frančula, 1971;Grafarend and Niermann, 1984). ...
... N. B. ln 2 (ℎ/ ) = ln 2 ( /ℎ), so no precautions are necessary to ensure that ℎ is the maximal rather than the minimal linear scale. The overall map distortion over a spherical surface can be measured effectively as the second moment (quadratic mean) of the local distortions, also known as the Airy-Kavrayskiy criterion (Frančula, 1971;Györffy, 2016;Kerkovits, 2020). Here, areal and angular distortions will be weighted with their undesirability and 1 − respectively: ...
Article
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... Pictures can be found at https://at-a-lanta.nl/weia/cupola.html. (Böhm, 2006;Canters, 2002;Eckert-Greifendorff, 1935;Frančula, 1971;Gall, 1885;Arno Peters, 1967;Aribert Peters 1978) ...
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... 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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... The Winkel Tripel map projection with the standard parallel at 50°28 ′ latitude was selected. Frančula (1971) has demonstrated the suitable properties of this map projection for world maps, and since 1998, the National Geographic Society has used it for world maps (http:// www.csiss.org/map-projections/microcam/mapnews. htm). ...
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Article
In map projections theory, various criteria have been proposed to evaluate the mean distortion of a map projection over a given area. Reports of studies are not comparable because researchers use different methods for estimating the deviation from the undistorted state. In this paper, statistical methods are extended to be used for averaging map projection distortions over an area. It turns out that the measure known as the Airy–Kavrayskiy criterion stands out as a simple statistical quantity making it a good candidate for standardization. The theoretical arguments are strengthened by a practical map projection optimization exercise.
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”.
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