Eckert-Greifendorff projection

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{aligned}x&=2\operatorname {laea} _{x}\left({\frac {\lambda }{4}},\varphi \right)\\y&={\tfrac {1}{2}}\operatorname {laea} _{y}\left({\frac {\lambda }{4}},\varphi \right)\end{aligned}}

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

{\begin{aligned}x&={\frac {4{\sqrt {2}}\cos \varphi \sin {\frac {\lambda }{4}}}{\sqrt {1+\cos \varphi \cos {\frac {\lambda }{4}}}}}\\y&={\frac {{\sqrt {2}}\sin \varphi }{\sqrt {1+\cos \varphi \cos {\frac {\lambda }{4}}}}}\end{aligned}}

The inverse is calculated with the intermediate variable

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

The longitude and latitudes can then be calculated by

{\begin{aligned}\lambda &=4\arctan {\frac {zx}{4\left(2z^{2}-1\right)}}\\\varphi &=\arcsin zy\end{aligned}}

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

^ Flattening the Earth: Two Thousand Years of Map Projections, John P. Snyder, 1993, pp. 130–133, ISBN 0-226-76747-7.
^ Weisstein, Eric W. "Hammer–Aitoff Equal-Area Projection." From MathWorld—A Wolfram Web Resource