Article

Problem of the Optimal Cartographic Projections

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Abstract

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.

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Thesis
Full-text available
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.
Chapter
In this memoir, J. H. Lambert for the first time formulates desired properties of maps from parts of the sphere to the plane such as to preserve angles or to preserve the proportion of area or to map great circles to lines or circles. He obtains conditions for such maps in terms of differential equations and constructs explicit families of examples some of which have become important tools in modern cartography.
Article
A redefinition of the North American geodetic networks may well require a reappraisal of the various plane coordinate systems in use in North America. Now “plane coordinate system” and “map projection” are really only different names for the same thing. An approach to coordinate conversion is described in which each map projection (plane coordinate system) is not analyzed as an individual, but the whole set of map projections is regarded as an integrated system. The various projections are displayed in a “family tree” and conversion between two coordinate systems is performed by following the most direct path in this tree. The advantages of this approach are that 1) reduction to latitude and longitude as an intermediate step is usually unnecessary and 2) the individual steps in the tree are usually quite simple mathematically owing to the flexibility in the parameters used. The traditional approach using latitude and longitude as essential parameters often leads to ugly mathematical functions best approximated by Taylor series whose coefficients may be quite awkward to determine.
Article
Chesterton did not, of course, intend this gibe to be taken literally. But the more we consider what he would doubtless have called the “Higher Geodetics”, the more we must conclude that there is some literal justification for it. Not only are straight lines straight. A sufficiently short part of a curved line may also be considered straight, provided that it is continuous (i.e. does not contain a sudden break or sharp corner), and provided we are not concerned with a measure of its curvature. Similarly a square mile or so on the curved surface of the conventionally spheroidal earth is to all intents and purposes flat. We shall achieve a considerable simplification, without any approximation, in the treatment of the present subject by getting back to these fundamental glimpses of the obvious, whether the formalists and conformalists accept them or not.
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