Aitoff projection

From Wikipedia, the free encyclopedia

An Aitoff projection of the Earth

The Aitoff projection is a modified azimuthal map projection. Proposed by David A. Aitoff in 1889, it is the equatorial form of the azimuthal equidistant projection, but stretched into a 2:1 ellipse while halfing the longitude from the central meridian:

$(\lambda, \phi)\mapsto\left(azeq_x(\lambda/2, \phi), azeq_y(\lambda/2, \phi)/2\right)$

where $azeq_x\,$ and $azeq_y\,$ are the x and y components of the equatorial azimuthal equidistant projection. Written it out explicitly, the projection is:

$(\lambda, \phi)\mapsto\left(2 z \cos \phi \sin (\lambda/2)/\sin z, z \sin \phi/\sin z\right)$

where

$z = \arccos(\cos \phi \cos (\lambda /2))\,$ .

These equations break down when $\lambda\,$ and $\phi\,$ are both zero, but $(\lambda, \phi)\mapsto(0,0)$ in this case. In all these formulas, $\lambda\,$ is the longitude from the central meridian and $\phi\,$ is the latitude.

Three years later, Ernst Hermann Heinrich Hammer suggested the use of the Lambert azimuthal equal-area projection in the same manner as Aitoff, producing the Hammer projection. While Hammer was careful to cite Aitoff, there has been some confusion since, wherein Aitoff has been attributed to Hammer's projection.^[1]