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Die vorteilhaftesten Abbildungen in der Atlaskartographie
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: ...
Seeking low distortion maps, it is usual to assume that areal and angular distortions are equally undesirable on the map. However, this might not be the case for certain map thematics. Should angular distortions be a bit less preferred to areal distortions, maps of unbalanced distortions may be developed. In this paper, the known analytic solution for the best cylindrical map projection is extended to such more general requirements by utilizing calculus of variations. The overall distortion of the resulted mappings are calculated and compared to each other to explore the distortion characteristics of these intentionally unbalanced map projections.
... 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) ...
In this article a new equal-area projection for small scale world maps is presented, with north-south asymmetry. The equations used form a system including well-known equivalent projections designed by Lambert, Hammer and Wagner. A new criterion is used to compare the quality of projections, concentrating on angular distortion on continents on the far side of the map. For Hammer's and Wagner's projection a version is presented which is having less distorted continents than the original versions. The final projection is named the Cupola Projection. It is having the Pacific Ocean at the far sides of the map and therefore less land close to the border of the map than would be the case with other central meridians. Due to its asymmetry landmasses near the borders of the map are less distorted than is the case with other frequently used equal-area world maps.
... 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: ...
Both the media and the geosciences often use small-scale world maps for demonstrating global phenomena. The most important demands on the projection of these maps are: (1) the map distortions have to be reduced as much as possible; (2) the outline shape of the mapped Earth must remind the viewer of the Globe. If the map theme to be illustrated requires neither equivalency (nor, which rarely happens, conformality) nor prescriptions for the map graticule, an aphylactic non-conical projection with simultaneously minimized angular and area distortions is advisable. In this paper, a graticule transformation by a parameterizable function helps to convert minimum distortion pointed-polar pseudocylindrical projections for world maps into general non-conical projections with further minimized distortions. The maximum curvature of the outline shape will be moderated at the same time in order to obtain a definitely pointed-polar character.
Cartographic projections are basis for the graphical representation of various territories in small scale mapping. Proper selection of projection reduces the deformation of the presented territory, which is bounded by a boundary line. In most cases, this border line is not a mathematically defined curve, which is most easily displayed in the form of a closed polygon. The optimal cartographic projections based on a selected criterion of quality are those whose constants lead to the smallest value of the criterion. In the presented work it is recommended to use Airy-Kavrajski criterion whose minimization is actually minimization of the second Euclidean norm. The solution of optimal projections of various classes is reduced to the method of least squares. Fast modern computers enable the optimization of an arbitrary territory by evaluating the selected criterion in a finite number of points.
Ocean maps are rarely in the scope of current studies on minimum-distortion map projec-tions. This study aims to create an uninterrupted map projection to display planet Earth as the Blue Planet: the aspect of the projection is rotated into the middle of the water surface; favourable map distortions are optimized numerically across the World Ocean. The paper starts with a short overview of existing similar projections. In the next pages, the reader may find the detailed description on the development of the new mapping. The paper concludes with maps and distortion analysis in the proposed projection and thoughts about its potential usefulness.
This paper presents the Cameron Aquatic Projection, a novel concept designed to depict the world’s surface hydrosphere as it is, a continuous unbroken unit including oceans and rivers. After a review of related past projections, the paper will describe the parameters of the projection, according to the existing layout of surface water on Planet Earth. This description of the design principles is followed by the mathematical realization of the proposal. The article concludes with completed examples of the projection and suggests future applications.
In map projection theory, it is usual to utilize numerical quadrature rules to estimate the overall map distortion. However, it is not known which method is the most efficient to approximate this integral. In this paper, overall map distortion is calculated analytically by a computer algebra system. Various integration methods are compared to the exact results. Some calculations are also performed on irregular spherical polygons. Considering the experiments, the author suggests utilizing the first-order Gaussian quadrature as it always gave reasonable results, although it is not the best for all cases.
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.
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