# Hammer projection

Hammer projection of the world

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 toward the outer limbs, 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:

${\displaystyle x={\mathrm {laea} }_{x}\left({\frac {\lambda }{2}},\phi \right)}$
${\displaystyle y={\frac {1}{2}}{\mathrm {laea} }_{y}\left({\frac {\lambda }{2}},\phi \right)}$

where ${\displaystyle {\mathrm {laea} }_{x}}$ and ${\displaystyle {\mathrm {laea} }_{y}}$ are the x and y components of the equatorial Lambert azimuthal equal-area projection. Written out explicitly:

${\displaystyle x={\frac {2{\sqrt {2}}\cos(\phi )\sin \left({\frac {\lambda }{2}}\right)}{\sqrt {1+\cos(\phi )\cos \left({\frac {\lambda }{2}}\right)}}}}$
${\displaystyle y={\frac {{\sqrt {2}}\sin(\phi )}{\sqrt {1+\cos(\phi )\cos \left({\frac {\lambda }{2}}\right)}}}}$

The inverse is calculated with the intermediate variable

${\displaystyle z\equiv {\sqrt {1-\left({\frac {1}{4}}x\right)^{2}-\left({\frac {1}{2}}y\right)^{2}}}}$

The longitude and latitudes can then be calculated by

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

where ${\displaystyle \lambda }$ is the longitude from the central meridian and ${\displaystyle \phi }$ is the latitude.[1][2]

Visually, the Aitoff and Hammer projections are very similar. The Hammer has seen more use because of its equal-area property. The Mollweide projection is another equal-area projection of similar aspect, though with straight parallels of latitude, unlike the Hammer's curved parallels.

### Briesemeister

William A. Briesemeister presented a variant of the Hammer in 1953. In this version, the central meridian is set to 10°E, the coordinate system is rotated to bring the 45°N parallel to the center, and the resulting map is squashed horizontally and reciprocally stretched vertically to achieve a 1.75:1.0 aspect ratio instead of the 2:1 of the Hammer. The purpose is to present the land masses more centrally and with lower distortion.[3]

### Nordic

Before projecting to Hammer, John Bartholomew rotated the coordinate system to bring the 45° north parallel to the center, leaving the prime meridian as the central meridian. He called this variant the "Nordic" projection.[3]