Eckert V and Eckert VI projection

World map in Eckert VI projection (Eckert V has the same external shape)

The Eckert V projection and the Eckert VI projection are two sinusoidal pseudo-cylindrical map projections developed by Max Eckert-Greifendorff and published in 1906. Eckert-VI in particular (can also be found as Kawraiski-VI ) found wider distribution.

properties

All Eckert projections are map network designs with parallel latitudes of unequal length , with the poles being shown in the form of a polar line that is half as long as the equator . The central meridian is also 1: 2 to the equator.

With these two variants, the longitudinal circles are shown as sine curves.

The Eckert-V 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-IV (ellipsoid) - 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 49 ° 16' N and S, within this area the map elements are stretched in an east-west direction (at Eckert-VI at the equator by 29 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 49 ° 16 ′ and the central meridian in the latter. The center of the map is quite undisturbed, especially at Eckert-V.

calculation

The Eckert V projection is basically the arithmetic mean of a sinusoidal projection and a flat map . For the Eckert VI projection, the widths are modified for equal area and the lengths are adjusted.

To describe the design mathematically, consider the earth's surface as a sphere with a radius and describe points on it using the spherical coordinates . The image of this point describes its coordinates , where the x-axis is the image of the equator and the y-axis is the image of the meridian at the angle : ${\ displaystyle R}$ ${\ displaystyle (\ varphi, \ lambda)}$ ${\ displaystyle (x, y)}$ ${\ displaystyle \ lambda = \ lambda _ {0}}$

Eckert-V:

{\ displaystyle x = R (\ lambda - \ lambda _ {0}) {\ frac {1+ \ cos {\ varphi}} {\ sqrt {2+ \ pi}}}}

{\ displaystyle \ approx 2 {,} 26750803 \, R (\ lambda - \ lambda _ {0}) (1+ \ cos {\ varphi})}

,

{\ displaystyle y = 2R {\ frac {\ varphi} {\ sqrt {2+ \ pi}}}}

{\ displaystyle \ approx 4 {,} 535016055 \, R \, \ varphi}

.

Eckert-VI:

{\ displaystyle x = R (\ lambda - \ lambda _ {0}) {\ frac {1+ \ cos {\ theta}} {\ sqrt {2+ \ pi}}}}

{\ displaystyle \ approx 2 {,} 26750803 \, R (\ lambda - \ lambda _ {0}) (1+ \ cos {\ theta})}

,

{\ displaystyle y = 2R {\ frac {\ theta} {\ sqrt {2+ \ pi}}}}

{\ displaystyle \ approx 4 {,} 535016055 \, R \, \ theta}

,

where is.

{\ displaystyle \ theta + \ sin {\ theta} = \ left (1 + {\ frac {\ pi} {2}} \ right) \ sin {\ varphi}}

{\ displaystyle \ approx 2 {,} 57079634 \, \ sin {\ varphi}}

As for the Eckert IV projection , the value of for Eckert VI is only given implicitly and can only be calculated approximately, for example using a Newton method . ${\ displaystyle \ theta}$

use

Zonobiome , on a slightly trimmed Eckert-VI

The projections are only useful for a world map . Since their smallest deviations are around 49 ° or 38 ° N / S and are on the central meridian or equator, they are particularly suitable for mapping countries in the middle latitudes in relation to the total surface. They have a particularly harmonious arrangement of the land masses, but were never really able to assert themselves, probably because of the "quite 'pointed' sine curve outer shape". 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 V projection would be the cheaper one, which shows the polar regions better.

The equal-area Eckert-VI was used, for example, in the Soviet Union for world distribution maps in the Atlas Mira (world atlas) from 1937 (in Russia the projection can be found as Kawraiski-VI , because it was published for it by Vladimir Kawraiski in 1936 ).

Further developments

Hölzel projection , rounded at the pole

A further development of the Eckert-V design is the Hölzel projection (Hölzel planisphere) . It was published by Ed. Hölzel for the Austrian Middle School Atlas (Konzenn Atlas) published in 1951 and is still widely used in this country today. This design rounds off the polar ice caps.

In 1992 Carlos A. Furuti published a variant with reduced sinus bends (to 57.29577951 ° / 90 °). This leads to a more "pleasing" exterior shape. This map network design is known as the Eckert-Furuti projection .

Similar designs

The Eckert-V and the Eckert-VI are similar to the Winkel-I-projection and the McBryde-Thomas-III-projection with slightly different external proportions - in fact, the Winkel-I is a generalization of the Eckert-V to pollines of any length, it is often used in a factor of 0.61; McBryde-Thomas-III uses ¹ ⁄ ₃ . The Wagner III and Wagner I projections , which use piecewise sinus curves, are similar. There are other similar designs from Kawraiski. The latter all have the advantage that they do not have to be iterated.

literature

Max Eckert: New designs for earth maps. In: Petermanns Mitteilungen, 1906 (52), 5, 97-109.
Max Eckert: The map science , 1921.

Web links

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

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

Individual evidence

↑ ^a ^b ^c Rolf Böhm: map projections - pseudocylindrical projections : Eckerts Erdkarteennetze , boehmwanderkarten.de (with illustrations, quotations ibid).

↑ ^a ^b Carlos Alberto Furuti: Flat-Polar Pseudocylindrical Projections: Six Projections by Eckert , progonos.com → Map Projection (accessed February 15, 2015).

↑ ^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-VI).

↑ ^a ^b ^c ^d ^e ^f ^g ^h ⁱ ^j ^k ^l John P. Snyder: An Album of Map Projections . USGS Professional Paper 1453. Denver 1989, ISBN 0-226-76747-7 , pp. 46 f. and 50 f . ( Web link to pdf , usgs.gov [accessed on February 11, 2015] Formulas p. 220, col. 2, 31–33).

↑ John P. Snyder: Flattening the Earth: Two Thousand Years of Map Projections . University of Chicago Press, 1997, ISBN 0-226-76747-7 , pp. 191 .

↑ ^a ^b Gerald I. Evenden: Cartographic Projection Procedures for the UNIX Environment - A User's Manual , p. 25 ( pdf ( page no longer available , search in web archives ) Info: The link was automatically marked as defective. Please check the link accordingly Instructions and then remove this note. , There p. 29; on remotesensing.org, ftp; with illustrations).@1@ 2

↑ ^a ^b ^c ^d ^e Eckert V , Eckert VI , arcgis.com.

↑ 1951 | A new beginning after the Second World War ( Memento of the original from February 7, 2015 in the Internet Archive ) Info: The archive link was automatically inserted and not yet checked. Please check the original and archive link according to the instructions and then remove this notice. , hoelzel.at → Large corporate atlas . @1@ 2

↑ Carlos A. Furuti: Uma Projeção Cartografica equivalents. Projeto A HAND. DCC | IMECC Unicamp, October 1992;
illustrated and discussed in Eckert's earth map networks, boehmwanderkarten.de (with formulas of the Furuti variant and a link to the work).

↑ Furuti: Some hybrid Pseudocylindrical Projections: Angle I , progonos.com ( Eckert V , see before).

[boehm-1] Rolf Böhm: map projections - pseudocylindrical projections : Eckerts Erdkarteennetze , boehmwanderkarten.de (with illustrations, quotations ibid).

[Furuti-2] Carlos Alberto Furuti: Flat-Polar Pseudocylindrical Projections: Six Projections by Eckert , progonos.com → Map Projection (accessed February 15, 2015).

[Snyder87-3] ↑ ^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-VI).

[Snyder89-4] ↑ ^a ^b ^c ^d ^e ^f ^g ^h ⁱ ^j ^k ^l John P. Snyder: An Album of Map Projections . USGS Professional Paper 1453. Denver 1989, ISBN 0-226-76747-7 , pp. 46 f. and 50 f . ( Web link to pdf , usgs.gov [accessed on February 11, 2015] Formulas p. 220, col. 2, 31–33).

[Snyder97-5] John P. Snyder: Flattening the Earth: Two Thousand Years of Map Projections . University of Chicago Press, 1997, ISBN 0-226-76747-7 , pp. 191 .