Aitoff projection

From Wikipedia, the free encyclopedia
Jump to: navigation, search
An Aitoff projection of the world

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:

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

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

x = \frac{2 \cos(\phi) \sin\left(\frac\lambda 2\right)}{\mathrm{sinc}(\alpha)}\,
y = \frac{\sin(\phi)}{\mathrm{sinc}(\alpha)}\,

where

\alpha = \arccos\left(\cos(\phi)\cos\left(\frac\lambda 2\right)\right)\,

and \mathrm{sinc}(\alpha) is the unnormalized sinc function with the discontinuity removed. In all of 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, some authors have mistakenly referred to the Hammer projection as the Aitoff projection.[1]

See also[edit]

References[edit]

  1. ^ Flattening the Earth: Two Thousand Years of Map Projections, John P. Snyder, 1993, pp.130-133, ISBN 0-226-76747-7.

External links[edit]