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The Hammer-Aitoff Projection

This map projection is an equal-area map projection which displays the world on an ellipse.

However, it is completely unlike the Mollweide projection. In the conventional case, the parallels are curved, and there is no stretching at the center of the map.

This map was created from a hemisphere of the equatorial case of Lambert's Azimuthal Equal-Area projection using a trick similar to the method by which the Miller Cylindrical projection was made from the Mercator projection.

This projection is also popular in the oblique case, producing maps like this:

Compare the Mollweide projection in precisely the same orientation:

The basic idea behind the Hammer-Aitoff projection is simple enough: first, place the whole world in one hemisphere by the simple expedient of dividing all longitudes by two, then to compensate for that, stretch the map out twice as wide.

Instead of using a factor of two, a factor of four was used to create the Eckert-Greifendorff projection:

Note that Adams' Equal-Area projection, which spaces the parallels as they are spaced on the central meridian of an equatorial Lambert's Azimuthal projection, could be thought of as the limiting case of this kind of manipulation, as the factor approaches infinity.

This projection was devised by Hammer, using the idea first used by Aitoff in making a compromise projection in such an ellipse from the Azimuthal Equidistant projection. Aitoff, however, then returned the compliment by using the basic method behind the Lagrange Conformal projection, but this time superimposing one equal-area cylindrical projection upon another. The ratio in scales was 9 to 10, and this time a compensating stretch was needed, to create an equal-area projection which I find quite appealing:

Since the same transformations that can be applied to the Hammer projection can also be applied to the Adams projection, one could further reduce shear by interrupting the modified Adams projection, and combining it with the Aitoff equal-area projection, somewhat after the manner used in the Oxford projection, to form a result like the following:

However, to make the pole lines shorter, to try to make the result a more balanced compromise between the Hammer-Aitoff on the one hand, and the cylindrical equal-area on the other, I used the sine of 75 degrees, or .9659258, as the factor to produce this projection:

However, that didn't seem to be an improvement, since the extra shearing seems to be worse than the stretching in the previous projection. However, in an oblique case of the projection:

it's possible to move all the important land masses out of the areas with high shearing, producing quite a pleasing result.

Also, if we reduce stretching at the cost of more shearing, we can then reduce shearing through the kind of interruptions tried above with Aitoff's projection, to which a projection such as the above, with a shorter pole line, might be more suited.

A projection of this type which is found as a world map on some atlases is the Wagner VII projection, also known as the Hammer-Wagner projection. The factor it uses is the sine of 65 degrees, or 0.906307787, for the superimposition of equal-area cylindrical projections. Simply using this factor (with, of course, inverse stretching of the map to keep the center of the map conformal) will produce the following map:

But the actual Hammer-Wagner projection, instead of being derived from an equal-area latitude transformation applied directly to Hammer's projection, moderates the compression of longitudes away from the prime meridian; instead of compressing the 360 degrees of longitude of the sphere into a hemisphere of Lambert's Azimuthal Equal-Area, with 180 degrees of longitude, it compresses the sphere further, into only 120 degrees of longitude, and then expanding it back, leading to a result like this:

But even that isn't quite the actual Wagner 7 or Hammer-Wagner projection. The projection is then stretched vertically, so that there are two conformal points on the prime meridian, located in temperate latitudes. I had some difficulty in calculating the factor of stretching, exclusive of that which compensates for the latitude and longitude transformations, which my mapping program is designed to automatically provide, that this projection uses, but I now believe it to be a factor of about 1.2650925, yielding this projection:

This equal-area projection has become quite popular for world maps in a number of recent atlases.

A paper by D. R. Steinwand, J. A. Hutchinson, and J. P. Snyder, in which an extensive analysis of shape distortion in equal-area projections was described, recommended four equal-area projections for use in statistical maps: the Goode Homolosine projection, the interrupted Mollweide, this projection, and the pseudocylindrical Wagner IV projection.

In the section on the Lagrange Conformal projection, I note that both that projection and the Hammer-Aitoff projection featured here provide a distribution of error that is perhaps very well suited to a number of countries which neither lack a long axis (like France) nor favor one to an extreme extent (like Chile), and, thus, are not best served by either an azimuthal projection on the one hand, or a cylindrical, conic, or polyconic projection on the other.

Here, then

is a map of Canada, drawn in the central part of an oblique Hammer-Aitoff projection centered on 55 degrees North latitude and 95 degrees West longitude.

For another example of how an oblique Hammer-Aitoff projection might be used for an area elongated in extent, here is a map of the Atlantic Ocean on such a projection:

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