Map Projections: Modified Azimuthal Projections

Map Projections

Three Modifications for Azimuthal Projections

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.

Original equatorial aspect of azimuthal equidistant map, inner hemisphere outlined

Inner hemisphere

Doubled longitudes

Doubled horizontal scale

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:

project the world with doubled longitudinal coordinates, effectively cramming everything into the inner hemisphere
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.

Projection equations follow directly from those for the equatorial azimuthal equidistant, substituting λ / 2 for λ and multiplying a factor 2 in x coordinates:

α = arccos (cos φ cos (λ/2))
x = 2αR cos φ sin (λ/2) / sin α
y = αR sin φ / sin α

Hammer's and Eckert-Greifendorff's Projections

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:

global scale is smaller than in Aitoff's
the projected inner hemisphere is not half as wide as the whole map, but encloses half its area
the final doubling restores scaled proportions and the final map is also equal-area
scale is no more constant along the major axes

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

Aitoff (top) and Hammer (bottom) maps at identical scales

Scales are different but overall lines are fairly similar in Aitoff and Hammer projections. Since differences in meridian spacing are hardly visible in the inner hemisphere, these two projections were 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 λ / 4 for λ / 2 and changing the x factor from 2 to 4.

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 the equidistant cylindrical projection, this time with Aitoff's. Again, φ₀ = ±arccos 2/π 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 conventional base parallels

α = arccos (cos φ cos (λ/2))
w = 0 if sin α = 0, 1 / sin α otherwise
x = R(λ cos φ₀ + 2wα cos φ sin (λ/2)) / 2
y = R(φ + wα sin φ) / 2