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[[File:Aitoff projection SW.jpg|300px|thumb|An Aitoff projection of the world]]
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 halving the longitude from the central meridian:

:<math>x = \mathrm{azeq}_x\left(\frac\lambda 2, \phi\right)\,</math>

:<math>y = \frac 1 2 \mathrm{azeq}_y \left(\frac\lambda 2, \phi \right)</math>

where <math>\mathrm{azeq}_x</math> and <math>\mathrm{azeq}_y</math> are the x and y components of the equatorial azimuthal equidistant projection.  Written out explicitly, the projection is:

:<math>x = \frac{2 \cos(\phi) \sin\left(\frac\lambda 2\right)}{\mathrm{sinc}(\alpha)}\,</math>

:<math>y = \frac{\sin(\phi)}{\mathrm{sinc}(\alpha)}\,</math>

where

:<math>\alpha = \arccos\left(\cos(\phi)\cos\left(\frac\lambda 2\right)\right)\,</math>

and <math>\mathrm{sinc}(\alpha)</math> is the unnormalized [[sinc function]] with the discontinuity removed. In all of these formulas, <math>\lambda</math> is the longitude from the central meridian and <math>\phi</math> 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, some authors have mistakenly referred to the Hammer projection as the Aitoff projection.<ref>''Flattening the Earth:  Two Thousand Years of Map Projections'', John P. Snyder, 1993, pp.130-133, ISBN 0-226-76747-7.</ref>

==See also==
{{Portal|Atlas}}
* [[List of map projections]]
* [[Mollweide projection]]
* [[Hammer projection]]

==References==
{{reflist}}

==External links==
* [http://www.radicalcartography.net/?projectionref Table of common projections]
* [http://www.uff.br/mapprojections/Aitoff_en.html An interactive Java Applet to study the metric deformations of the Aitoff Projection].
* [http://map.gsfc.nasa.gov/ Wilkinson Microwave Anisotropy Probe (WMAP)]

{{Map Projections}}

[[Category:Cartographic projections]]