Eckert III and Eckert IV projection

World map in Eckert IV projection

The Eckert III projection and Eckert IV projection are two of Max Eckert-Greifendorff developed and 1906 published ellipsoidal -area pseudo-cylindrical map projection with an oval outer shape. The Eckert IV design was particularly widespread in mid-20th century America .

properties

All Eckert projections are map network designs with parallel latitudes of unequal length , with the poles being shown as a line ( polar line ) that is half as long as the equator . The central meridian is also 1: 2 to the equator. The projection thus creates a well-proportioned format of the world map with an appealing overall picture and good orientation at the same time. The pole problem of the rectangular maps (the pole regions are either excessively wide or are becoming increasingly illegible) is solved by the design with the harmoniously dimensioned pole line.

In these two variants, the longitudes are shown as semi- ellipses or parts thereof. The two outermost longitudinal circles become semicircles .

The Eckert-III variant is neither true-angle (conformal) nor true-area , but has equidistant circles of latitude. Eckert-IV is - like the projections Eckert-II (straight) and Eckert-VI (sinusoid) - the equal-area version. In the first draft the scale is correct in the areas 37 ° 55 ′ N and S (if the total area is correct), in the second in the areas 40 ° 30 ′ N and S, within this area the map elements are stretched in an east-west direction (at Eckert-IV at the equator by 40 percent), outside compressed in a north-south direction. No point on the map is distortion-free, but the equator is angled in the former, the latitude 40 ° 30 ′ and the central meridian in the latter. The center of the map is quite undisturbed, especially with the Eckert-III.

calculation

If the radius of a sphere (whose surface serves as a model for the earth's surface), the central meridian and a point with the polar coordinates are given, the coordinates and the image point on the map can be calculated using the following formulas: ${\displaystyle R}$ ${\displaystyle \lambda _{0}}$ ${\displaystyle (\varphi ,\lambda )}$ ${\displaystyle x}$ ${\displaystyle y}$

Eckert III:

${\displaystyle x=2R\,(\lambda -\lambda _{0}){\frac {1+{\sqrt {1-({\frac {2\varphi }{\pi }})^{2}}}}{\sqrt {4\pi +\pi ^{2}}}}}$ ${\displaystyle \approx 0{,}4222382\,R\,(\lambda -\lambda _{0})\left(1+{\sqrt {1-{0},4052847\varphi ^{2}}}\right),}$

${\displaystyle y=4R{\frac {\varphi }{\sqrt {4+\pi }}}\approx 1{,}49679545\,R\varphi .}$

Eckert-IV:

${\displaystyle x=2R\,(\lambda -\lambda _{0}){\frac {(1+\cos \theta )}{\sqrt {4\pi +\pi ^{2}}}}}$ ${\displaystyle \approx 0{,}4222382\,R\,(\lambda -\lambda _{0})(1+\cos \theta ),}$

${\displaystyle y=2{\sqrt {\pi }}R{\frac {\sin \theta }{\sqrt {4+\pi }}}\approx 1{,}3265004\,R\sin \theta ,}$

where is. ${\displaystyle \theta +\sin \theta \cos \theta +2\sin \theta =\left(2+{\frac {\pi }{2}}\right)\sin \varphi }$ ${\displaystyle \approx {3},5707963\,\sin \varphi }$

As for the Eckert VI projection , the value for Eckert IV is only given implicitly. For example, the equation for can be solved using Newton's method . ${\displaystyle \theta }$

Conversely, if the coordinates are given on the map, the corresponding point on the spherical surface for Eckert-IV can be calculated as follows: ${\displaystyle x,y}$

${\displaystyle \theta =\arcsin \left[y{\frac {\sqrt {4+\pi }}{2{\sqrt {\pi }}R}}\right]}$ ${\displaystyle \approx \arcsin \left[{\frac {y}{1{,}3265004\,R}}\right],}$

${\displaystyle \varphi =\arcsin \left[{\frac {\theta +\sin \theta \cos \theta +2\sin \theta }{2+{\frac {\pi }{2}}}}\right]}$ ${\displaystyle \approx \arcsin(0,2800496(\theta +\sin \theta \cos \theta +2\sin \theta )),}$

${\displaystyle \lambda =\lambda _{0}+x{\frac {\sqrt {4\pi +\pi ^{2}}}{2R(1+\cos \theta )}}\approx \lambda _{0}+{\frac {x}{0{,}4222382\,R\,(1+\cos \theta )}}.}$

use

EGM96 - Geoid heights (gravitational anomalies), on an Eckert IV

The two projections are only useful for a world map . Since their smallest deviations are around 40 ° N / S and are on the central meridian, they are particularly suitable for setting countries in the middle latitudes in relation to the total surface area. Like all maps with parallel parallels, they are particularly suitable for zone models, for example for climatological, biological and similar thematic maps. Here the equally spaced Eckert III projection would be the cheaper one, which shows the polar regions better, but it is quite rare.

The Eckert IV equal-area projection is one of the most widely used of Eckert's map designs. Between 1940 and 1960 she was the third most in school textbooks from the United States card design used (after Goode Homolosine and the sinusoidal projection ).

Further developments

Since the draft constitutes a transcendent equation, the zeros of which can only be found iteratively, Karlheinz Wagner proposed two alternative formulas in 1949 that can be resolved algebraically. The first approximation tries to “adapt the Eckert IV draft as authentically as possible”, the other allows two parallel circles to be reproduced true to length of your choice .

Similar designs

A. Ortelius: Typus orbis Terrarum . From: Theatrum Orbis Terrarum , 1571

A map similar to Eckert-III was developed by Abraham Ortelius and used for the Typus orbis Terrarum  (No. 1) of his atlas Theatrum Orbis Terrarum , first printed in Antwerp in 1570. It is often called Eckert because it also has a poll line of ¹ ⁄ ₂ , but is actually a variation of an Appian projection , with ellipsoidal meridians running to a polar point for the inner hemisphere and circles of equal size for the other. This figure is called the Ortelius oval projection .

The Eckert VI is particularly similar to the Wagner IV projection (Putnin P2 ').

The advantage over the - overall ellipsoid - Mollweide and Hammer-Aitov projections is the more detailed representation of the middle latitudes. There are also similar designs that form the arithmetic mean between Mollweide and Plattkarte.

See also

• Eckert projection (overview)

literature

• Max Eckert: New designs for earth maps. In: Petermann's communications. 52, No. 5, 1906, pp. 97-109.
• Max Eckert: The map science , 1921.

Web links

Commons : Maps with Eckert IV projection  - collection of images, videos and audio files

• Eckert IV projection , Mathworld

Individual evidence

1. ↑ ^{a b} Rolf Böhm: map projections - pseudocylindrical projections : Eckerts Erdkarteennetze , boehmwanderkarten.de (with illustrations, quotations ibid).
2. ↑ ^{a b} Carlos Alberto Furuti: Flat-Polar Pseudocylindrical Projections: Six Projections by Eckert , progonos.com → Map Projection (accessed February 15, 2015).
3. ↑ ^{a b c d e f g} John P. Snyder: Map Projections - A Working Manual . USGS Professional Paper 1395. Denver 1987, ISBN 0-226-76747-7 , pp. 253-258 ( web link to pdf , usgs.gov [accessed July 24, 2013] with a more detailed history section and formulas for Eckart-IV).
4. ↑ ^{a b c d e f g h} John P. Snyder: An Album of Map Projections . USGS Professional Paper 1453. Denver 1989, ISBN 0-226-76747-7 , pp. 60 f . ( Web link to pdf , usgs.gov [accessed on February 11, 2015] Formulas p. 221, col. 1, 40–42).
5. ↑ John P. Snyder: Flattening the Earth: Two Thousand Years of Map Projections . University of Chicago Press, 1997, ISBN 0-226-76747-7 , pp. 191 .
6. ↑ ^{a b} Gerald I. Evenden: Cartographic Projection Procedures for the UNIX Environment - A User's Manual , p. 24 ( doi: https://doi.org/10.3133/ofr90284 ).
7. ↑ ^{a b c d} Eckert III , Eckert IV , arcgis.com.
8. ↑ FKC Wong: World map projections in the United States from 1940 to 1960. MA thesis, Syracuse University, Syracuse NY 1965, p. 101 (according to Snyder 1987, p. 253).
9. ↑ With open cards - The cards of others , on arte.tv, accessed on December 1, 2013.
10. ^ ^{A b} Karlheinz Wagner: Cartographic network drafts , 1st edition Leipzig 1949, p. 222; Wagner did not give these creations a name;
    illustrated and discussed in Eckerts Erdkarteennetze , boehmwanderkarten.de;
    for the book see map projections - Wagner's world map networks, boehmwanderkarten.de.
11. ↑ CA Furuti: Oval and Extended Globular Maps , progonos.com;
    Map projections - globular projections: Ortelius Oval , boehmwanderkarten.de;
    Ortelius oval , mapthematics.com.