Eckert-Greifendorff projection

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:

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

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

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

The inverse is calculated with the intermediate variable

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

The longitude and latitudes can then be calculated by

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

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

• List of map projections
• Hammer projection
• Eckert projection