The equatorial aspect of the azimuthal equidistant projection presents the whole world in the familiar “horizontal” aspect; however, there is significant areal exaggeration near the map boundaries.
Noticing that an azimuthal equidistant map encloses an “inner” hemisphere in a disc whose radius is half that of the whole map, Aitoff proposed a very simple, yet attractive modification:
The resulting projection, no more azimuthal, is equidistant only along the Equator and central meridian.
Forward projection equations follow directly from those for the equatorial azimuthal equidistant, substituting for and multiplying a factor 2 in the abscissas:
Aitoff's approach had been pioneered by Johann Lambert, with the compression of the azimuthal stereographic leading to the “Lagrange” projection.
Aitoff's work was itself modified by Hammer, whose projection applied the same idea, but to Lambert's azimuthal equal-area projection instead. As a consequence:
Again, formulas can be deduced replacing by , this time in Lambert's equations:
Scales are different but overall lines are fairly similar in Aitoff and Hammer projections. Differences in the graticule spacing are hardly visible in the inner hemisphere, and these two projections have been frequently mislabeled.
Hammer's design was in turn modified by Eckert-Greifendorff, in a projection applying a further 2 : 1 rescaling. Therefore equations are identical, except for substituting for and changing the factor from 2 to 4.
The limiting case for equal-area projections based on compressing longitudes and expanding abscissas by reciprocal factors, keeping the central meridian's scale constant, is the quartic authalic, a pseudocylindrical projection.
Yet another modification of Aitoff's projection was devised by Winkel. Much like in his first and second hybrid maps, his tripel projection averages and -coordinates of the equidistant cylindrical projection, this time with the Aitoff projection's. Again,
are usually chosen as
parallels for the cylindrical base (although the final
projection has no standard parallels).
Equations follow directly from Aitoff's and the equidistant cylindrical's: