Hammer projection

Hammer projection of the world.

The Hammer projection is an equal-area map projection, described by Ernst Hammer in 1892. 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.

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.

See also

• Mollweide projection
• Aitoff projection

External links

• Table of common projections
• An interactive Java Applet to study the metric deformations of the Hammer-Aitoff Projection.