# 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:

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

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

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

The inverse is calculated with the intermediate variable

$z \equiv \sqrt{1 - \left(\frac1 4 x\right)^2 - \left(\frac1 2 y\right)^2}$

The longitude and latitudes can then be calculated by

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

where $\lambda$ is the longitude from the central meridian and $\phi$ 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]