

Inner hemisphere  Doubled longitudes  
Doubled horizontal scale: Aitoff's projection 
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:
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 (top) and Hammer (bottom) maps at identical scales 
Aitoff's work was itself modified by Hammer, whose projection applied the same idea, but to Lambert's azimuthal equalarea 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 EckertGreifendorff, 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 equalarea 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.
Flattening parallels 

Modified equatorial azimuthal equalarea maps with reciprocal factors for longitude compression/horizontal expansion. 
Winkel tripel map with conventional (top) and 40° reference parallels 