Eckert-Greifendorff projection explained

The Eckert-Greifendorff projection is an equal-area map projection described by Max Eckert-Greifendorff in 1935. Unlike his previous six projections, it is not pseudocylindrical.

Development

Directly inspired by the Hammer projection, Eckert-Greifendorff suggested the use of the equatorial form of the Lambert azimuthal equal-area projection instead of Aitoff's use of the azimuthal equidistant projection:

\begin{align} x & = 2 \operatorname{laea} _{x\left( (λ)/(4)} , \varphi\right) \\ y & = \tfrac 12 \operatorname{laea} _{y\left( (λ)/(4)} , \varphi\right) \end{align}

where laea and laea are the x and y components of the equatorial Lambert azimuthal equal-area projection. Written out explicitly:

\begin{align} x & = (4 \sqrt 2 \cos \varphi \sin (λ)/(4))/(\sqrt{ 1 + \cos \varphi \cos (λ)/(4)) } \\y &= \frac \end

The inverse is calculated with the intermediate variable

z \equiv \sqrt{ 1 - \left(\tfrac 1 { 16 } x\right) ^{2} - \left(\tfrac 12 y\right) ^{2}}

The longitude and latitudes can then be calculated by

\begin{align} λ & = 4 \arctan (zx)/(4 \left( 2 z ^{2} -1 \right)) \\ \varphi & = \arcsin zy \end{align}

where λ is the longitude from the central meridian and φ is the latitude.^{[1] [2]}

See also

• List of map projections
• Hammer projection
• Eckert projection

Notes and References

1. Flattening the Earth: Two Thousand Years of Map Projections, John P. Snyder, 1993, pp. 130–133, .
2. Weisstein, Eric W. "Hammer–Aitoff Equal-Area Projection." From MathWorld—A Wolfram Web Resource