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:

$\begin{array}{rl}x& =2{\mathrm{laea}}_x(\frac{\lambda }{4},\phi )\\ y& =\frac{1}{2}{\mathrm{laea}}_y(\frac{\lambda }{4},\phi )\end{array}$

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

$\begin{array}{rl}x& =\frac{4\sqrt{2}\cos \phi \sin \frac{\lambda }{4}}{\sqrt{1+\cos \phi \cos \frac{\lambda }{4}}}\\ y& =\frac{\sqrt{2}\sin \phi }{\sqrt{1+\cos \phi \cos \frac{\lambda }{4}}}\end{array}$

The inverse is calculated with the intermediate variable

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

The longitude and latitudes can then be calculated by

$\begin{array}{rl}\lambda & =4\arctan \frac{zx}{4(2z^2-1)}\\ \phi & =\arcsin zy\end{array}$

where λ is the longitude from the central meridian and φ is the latitude.

See also

• List of map projections
• Hammer projection
• Eckert projection