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Three Modifications for Azimuthal Projections

Equatorial azimuthal equidistant projection with inner hemisphere highlighted
Ordinary equatorial azimuthal equidistant map, with inner hemisphere highlighted
Inner hemisphere of equatorial polar azimuthal equidistant projection Modified equatorial azimuthal equidistant projection with compressed longitudes
Inner hemisphere Doubled longitudes
Aitoff's projection with inner hemisphere highlighted
Doubled horizontal scale: Aitoff's projection

Deducing 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:

  1. project the world with doubled longitudinal coordinates, effectively cramming everything into the inner hemisphere
  2. double the horizontal scale, stretching the disc into a 2:1 ellipse
The resulting projection, no more azimuthal, is equidistant only along the Equator and central meridian.
Compressing longitudes
Different compression factors applied to an equatorial azimuthal map
This series of modified equatorial azimuthal equidistant maps shows how compressing longitudes by a factor from 1 to 2 puts the whole world into the space previously occupied by a single hemisphere.

Forward projection equations follow directly from those for the equatorial azimuthal equidistant, substituting lambda/2 for lambda and multiplying a factor 2 in the abscissas:

alpha=acos(cos phi cos (lambda/2)); k=0 if alpha=0, alpha R/sin alpha otherwise;x=2k cos phi sin (lambda/2); y = k sin phi

Aitoff's approach had been pioneered by Johann Lambert, with the compression of the azimuthal stereographic leading to the "Lagrange" projection.

Deriving Hammer's and Eckert-Greifendorff's Projections

Combined maps with Aitoff (top) and Hammer (bottom) projections
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 equal-area projection instead.  As a consequence:

Again, formulas can be deduced replacing lambda by lambda/2, this time in Lambert's equations:

k=R sqrt(2/(1+cos phi cos (lambda/2)));x=2k cos phi sin (lambda/2); y=k sin phi

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 lambda/4 for lambda/2 and changing the x 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.

Flattening parallels
Modified equatorial equal-area azimuthal maps
Modified equatorial azimuthal equal-area maps with reciprocal factors for longitude compression/horizontal expansion.

Winkel's Tripel 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 x and y-coordinates of the equidistant cylindrical projection, this time with the Aitoff projection's. Again,
are usually chosen as the standard parallels for the cylindrical base (although the final projection has no standard parallels).
Equations follow directly from Aitoff's and the equidistant cylindrical's:
Winkel tripel map with two different reference parallels
Winkel tripel map with conventional (top) and 40° reference parallels
alpha=arccos(cos phi cos(lambda/2));w=0 if sin alpha=0,1/sin alpha otherwise;x=R(lambda cos phi0+2w alpha cos phi sin(lambda/2))/2;y=R(phi+w alpha sin phi)/2

HomeSite MapEquidistant Cylindrical, Winkel I/IIHow Projections are CreatedAzimuthal Projections    February 24, 2014
Copyright © 2001 Carlos A. Furuti