# Hammer projection

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:

\displaystyle{ \begin{align} x &= \operatorname{laea}_x\left(\frac{\lambda}{2}, \varphi\right) \\ y &= \tfrac12 \operatorname{laea}_y\left(\frac{\lambda}{2}, \varphi\right) \end{align} }

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

\displaystyle{ \begin{align} 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{align} }

The inverse is calculated with the intermediate variable

$\displaystyle{ z \equiv \sqrt{1 - \left(\tfrac14 x\right)^2 - \left(\tfrac12 y\right)^2} }$

The longitude and latitudes can then be calculated by

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

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

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 7:4 aspect ratio instead of the 2:1 of the Hammer. The purpose is to present the land masses more centrally and with lower distortion.

### 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.