32. ECKERT IV AND VI PROJECTIONS #
SUMMARY #
- Pseudocylindrical.
- Equal-area.
- Central meridian is a straight line; 180th meridians of Eckert IV are semicircles; all other meridians are equally spaced elliptical arcs on Eckert IV and sinusoidal curves on Eckert VI.
- Parallels are unequally spaced straight lines, parallel to each other. Poles are straight lines half as long as the Equator.
- Scale is true along latitudes 40°30’ N. and S. on Eckert IV and 49°16’ on Eckert VI.
- Used for world maps.
- Presented by Eckert in 1906.
HISTORY AND USAGE #
In 1906 Max Eckert (1868- 1938) of Kiel, Germany, presented a set of six new projections in which all the poles are lines half as long as the Equator, and in which all parallels of latitude are straight lines parallel to each other. The central meridian on each is also half the length of the Equator (Eckert, 1906). Numbers 4 and 6 are of most significance and are discussed here in detail.
Of the six projections, nos. 2, 4, and 6 are equal-area, and nos. 1, 3, and 5 are not equal-area but have equally spaced parallels. For nos. 1 and 2, the meridians are straight lines broken at the Equator, and those projections are therefore little more than novelties with graticules composed entirely of straight lines, but with unnecessary distortion especially at the Equator. The meridians on nos. 3 and 4 are elliptical arcs, while on 5 and 6 they are sinusoidal curves, with the exception of the straight central meridians, and (on 3 and 4) semicircular outer meridians.
No. 3, with equidistant parallels and elliptical arcs has occasionally been identified as the same as the Ortelius projection, named for the famous cartographer Abraham Ortelius who used a somewhat similar projection in 1570 for his world map. The border, the central meridian, and the parallels of the two projections are shown almost identically, and the meridians on each are equally spaced along the Equator. The shapes of most meridians, however, are different. On the Ortelius, they are circular arcs, semicircles for meridians at or more than 90° from the central meridian, and circular arcs intersecting the central meridian at the poles within 90° of the central meridian.
FIGURE 56.—Eckert IV projection. An equal-area pseudocylindrical with poles half the length of Equator. Outer meridians are semicircles; others are elliptical arcs. FIGURE 57.— Eckert VI projection. Like figure 56, this is an equal-area pseudocylindrical projection with poles half the length of Equator. The meridians, however, are sinusoidal curves.
FEATURES #
The Eckert IV projection is bounded by two semicircles representing the 180th meridians and two straight lines connecting the ends of the semicircles. These straight lines represent the two poles, which are half the length of the Equator. Meridians are equally spaced semiellipses ranging in eccentricity from zero for the outer circular meridians to 1 for the straight central meridians. The parallels are straight lines parallel to the Equator and spaced to provide correct area within the border. They are therefore unequally spaced and closer together near the poles. There is a north-south stretching of shape at the Equator amounting to 40 percent relative to east-west dimensions. This stretching decreases along the central meridian to zero at latitudes 40°30’ N. and S. and becomes flattening beyond these parallels. The scale is correct only along these two parallels, and the only points free of distortion are at the intersection of these two points with the central meridian. Nearer the poles, the geographical features of the map are flattened in a north-south direction.
The Eckert VI projection of the world is bounded by two sinusoidal curves which have the same shape as the 90th meridians of the Sinusoidal projection. Like the border of the Eckert IV, these curved meridians are connected with two straight lines connecting the ends of the curves. These straight lines, half the length of the Equator, are the poles. The other meridians are equally spaced sinusoids, except for the straight central meridian, and the other parallels are straight and parallel to each other. To preserve area, the parallels must be unequally spaced, farther apart at the Equator than at the poles. As a result, there is a 29 percent north-south stretch at the Equator, relative to east-west dimensions. Other general comments concerning distortion of the Eckert IV apply to Eckert VI, but the latitudes of true scale are 49°16’ N. and S.
These projections are for world maps, not regional maps, and there is no need for the ellipsoidal forms.
FORMULAS FOR THE SPHERE #
The forward formulas for both Eckert IV and Eckert VI require iteration. Given $R, \lambda_0, \phi,$ and $\lambda$, to find $x$ and $y$ (see Eckert IV numerical examples and Eckert VI numerical examples):
Eckert IV:
Eckert IV:
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TABLE 43.—Eckert IV and VI projections: Rectangular coordinates for 90th meridian