The Hammer projection is an equal-area map projection described by Ernst Hammer in 1892. Using the same 2:1 elliptical outer shape as the Mollweide projection, Hammer intended to reduce distortion in the regions of the outer meridians, where it is extreme in the Mollweide.
Development Directly inspired by the Aitoff projection, Hammer suggested the use of the equatorial form of the Lambert azimuthal equal-area projection instead of Aitoff's use of the azimuthal equidistant projection:
x = laea x ( λ 2 , φ ) y = 1 2 laea y ( λ 2 , φ ) {\displaystyle {\begin{aligned}x&=\operatorname {laea} _{x}\left({\frac {\lambda }{2}},\varphi \right)\\y&={\tfrac {1}{2}}\operatorname {laea} _{y}\left({\frac {\lambda }{2}},\varphi \right)\end{aligned}}}
where laeax and laeay are the x and y components of the equatorial Lambert azimuthal equal-area projection. Written out explicitly:
x = 2 2 cos φ sin λ 2 1 + cos φ cos λ 2 y = 2 sin φ 1 + cos φ cos λ 2 {\displaystyle {\begin{aligned}x&={\frac {2{\sqrt {2}}\cos \varphi \sin {\frac {\lambda }{2}}}{\sqrt {1+\cos \varphi \cos {\frac {\lambda }{2}}}}}\\y&={\frac {{\sqrt {2}}\sin \varphi }{\sqrt {1+\cos \varphi \cos {\frac {\lambda }{2}}}}}\end{aligned}}}
The inverse is calculated with the intermediate variable
z ≡ 1 − ( 1 4 x ) 2 − ( 1 2 y ) 2 {\displaystyle z\equiv {\sqrt {1-\left({\tfrac {1}{4}}x\right)^{2}-\left({\tfrac {1}{2}}y\right)^{2}}}}
The longitude and latitudes can then be calculated by
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