Aitoff projection

The Aitoff projection is a modified azimuthal map projection proposed by David A. Aitoff in 1889. Based on the equatorial form of the azimuthal equidistant projection , Aitoff first halves longitudes, then projects according to the azimuthal equidistant, and then stretches the result horizontally into a 2:1 ellipse to compensate for having halved the longitudes.

Expressed simply:

${\displaystyle x=2\operatorname {azeq} _{x}\left({\frac {\lambda }{2}},\varphi \right),\qquad y=\operatorname {azeq} _{y}\left({\frac {\lambda }{2}},\varphi \right)}$

where azeq _x and azeq _y are the x and y components of the equatorial azimuthal equidistant projection. Written out explicitly, the projection is:

${\displaystyle x={\frac {2\cos \varphi \sin {\frac {\lambda }{2}}}{\operatorname {sinc} \alpha }},\qquad y={\frac {\sin \varphi }{\operatorname {sinc} \alpha }}}$

where

${\displaystyle \alpha =\arccos \left(\cos \varphi \cos {\frac {\lambda }{2}}\right)\,}$

and sinc  α is the unnormalized sinc function with the discontinuity removed. In all of these formulas, λ is the longitude from the central meridian and φ 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. ^[1]

See also [ edit ]

• List of map projections
• Mollweide projection
• Hammer projection

References [ edit ]

External links [ edit ]

Wikimedia Commons has media related to Aitoff projection .

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