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August's Conformal Projection of the Sphere on a Two-Cusped Epicycloid

This projection is not one that you are likely to see often.

But it is a very nice projection. It is conformal, but unlike the Stereographic projection or Mercator's projection, it is possible to fit all the world in it, without any parts going to infinity.

Lagrange's projection, on the previous page, does do that, but the poles, although present, are the two points at which the projection ceases to be conformal. This projection does not even have that problem.

As a result, this projection was used as the basis for a famous map of ocean currents that you may have run across, and which is perhaps the only place many people have seen this projection. The illustration above shows, at least approximately, the particular transverse form of this projection used for that map. It first appeared in a paper, "Maps of the Whole World Ocean", by Athelstan Spilhaus, in a 1942 issue of the Geographical Review.

(Dr. Spilhaus continued working on mapping the oceans of the world continuously on a single map after that; this page shows a projection he designed for that purpose in 1980, or, perhaps, in 1979, as this site, containing an excellent booklet with numerous illustrations of map projections, states. He passed away at age 86 on March 29, 1998.)

Of course, one can place the world on this projection in many orientations.

This map is one attempt to place the world in a useful orientation on this projection. Note, though, that while Australia is placed in a part of the map where it is very much enlarged, its general shape is less distorted than that of parts of Asia, where the scale is smaller, but changes more rapidly.

Here is another version of this projection, this time in an oblique case rather than a transverse case:

this is in fact the same orientation that I used for maps in the Mollweide projection and the Hammer-Aitoff projection.

How is this projection made? The first step is to take the world, and project it using Lagrange's projection, so that the world is conformally mapped to a circle.

Then, the circle is conformally mapped to the inside of the epicycloid that forms the boundary of the map using complex numbers. Functions over the complex numbers are an inexhaustible source of conformal mappings, since functions are extended to complex numbers in the fashion that permits them to be differentiable over the complex plane, and the condition for differentiability and for forming a conformal mapping are one and the same. Of course, one can form many useless conformal mappings into weird shapes that way, but many very useful conformal maps have been found using complex numbers.

The Lagrange projection of the globe is placed on the complex plane so that the north pole is at z=1, and the south pole is at z=-1. Then, the function 3z-z^3, which has inflection points at z=-1 and z=1 along the real line (places where it changes from increasing to decreasing or back again, like the sine function at 90 degrees and 270 degrees) is used to map the Lagrange projection to August's projection. It is because of the inflection points that one gets the cusps in the bounding curve; one could use the sine function instead to make a projection that looked somewhat like August's projection.

This can be illustrated by the following image, in which a map of the world on Lagrange's conformal projection is superimposed on a gradient diagram of the function 3z-z^3. The three zeroes of this cubic polynomial are clearly visible on the real line as areas bounded by approximate circles, and the fact that angles are doubled at the inflection points can also be seen.

The conformal transformation involved can be seen in action by means of the following diagram, containing a gradient diagram of the identity function (with the axes on a larger scale) on which a map in August's conformal projection is superimposed:

corresponding areas on the maps in the two images are the same color within the gradient diagram. The North Pole of the Lagrange projection is on the point 1+0i in the first diagram, where 3z-z^3 evaluates to 2; in the second diagram, the North Pole of August's Conformal Projection is on the point 2+0i, where the identity function, z itself, also evaluates to 2, and hence both points are on the corner between the same four colors (which are in two groups of two colors which do not contrast very well; that there are four colors and not two is perhaps a bit easier to see around the South Pole) in the gradient diagram, which bound the areas with amplitude from 1 to 2 and 2 to 3, and phase from 330° to 360° and from 0° to 30°.

The behavior of the inflection points for complex arguments is also important, because the cusps could easily wrap around for other functions. Note, of course, that for the sine function, the circle with radius pi/2 is used instead of the unit circle.

And the above is the projection I first produced on my Commodore 64 computer before I was able to determine the actual formula for August's Conformal projection. This projection has a close resemblance to the Eisenlohr, but it is not the same as that projection.

This projection appears to have greater size distortion than August's conformal, and the Eisenlohr would appear to be even slightly worse, but in fact the Eisenlohr, because it has a uniform scale everywhere along its boundary, has the minimal overall distortion of size.

Here is the actual Eisenlohr projection itself:

Why are appearances decieving? Although August's conformal projection has greater size distortion near the poles, it has less distortion for areas on the Equator far from the central meridian. For world maps in conventional aspect, this will appear to be more important, but for applications where every part of the globe is equally important, the Eisenlohr would be advantageous, despite being somewhat more complicated to calculate.

As I had some difficulty in implementing the Eisenlohr, despite having located equations for it, and as information about the projection is difficult to locate on-line, it may be helpful to provide the equations for this projection, essentially as they appeared in "An Album of Map Projections" by Snyder and Voxland (USGS Professional Paper 1453), but also confirmed by my having finally successfully coded the projection:

S1 = sin ( ---- )

C1 = cos ( ---- )

                      sin ( --- )
T = ----------------------------------------------
           lat                 /  cos( lat )
     cos ( --- ) + 2 * C1 *   /  ------------
            2               \/        2

        /     2
C =    / ------------
      /          2
    \/    ( 1 + T  )

             /                                 _______________
            /        lat                      /  cos( lat )
           /   cos ( --- ) + ( C1 + S1 ) *   /  ------------
          /           2                    \/         2
V =      /    -------------------------------------------------
        /                                      _______________
       /             lat                      /  cos( lat )
      /        cos ( --- ) + ( C1 - S1 ) *   /  ------------
    \/                2                    \/         2

x = C * ( V - --- ) - 2 ln( V )

y = C * T * ( V + --- ) - 2 atan( T )

X and Y are scaled up by a factor of 3 plus the square root of 8 in the equations as given, omitted above for simplicity; but this causes X to range between plus and minus 2 times pi; so, for my program, where I scale everything to being from -180 to +180, I multiply by 90/pi, and use 166.9721 as my scale factor.

Incidentally, the two terms in the expansions of x and y are each conformal projections in themselves; x = ln(V), y = atan(T) produces a conformal projection in a shape resembling an ellipse with a limited elongation. (However, the shape is unlikely to actually be an ellipse; elliptic integrals, as used for Guyou's projection, to be seen next, are required for that.)

The reason the Eisenlohr projection is optimal is because the scale is uniform on its boundary; a conformal projection from one region to another is optimal when the condition of uniform boundary scale is met. But there is no reason why the world has to be conformally projected on any particular shape. Thus, shortening the extent of the link between the two hemispheres might improve the projection, since two Stereographic hemispheres have a lower distortion than a conformal map of the whole world.

One might map a polar Stereographic projection onto one with a smaller scale, such that its equator coincides with 45 degrees north latitude, and then place the pole of the result in the center of a map using the cosine-based projection above to attempt to move towards the lower distortion of two Stereographic hemispheres by reducing the connection between the two halves of the projection. The following

is an oblique case of this projection, tilted five degrees backwards along the prime meridian. Instead of the center being at a smaller scale, it is now at a larger scale than the rest of the map; a further transformation to improve the map by making the scale more closely constant should be possible, and this would provide an interesting conformal projection.

If we use August's projection, and reduce the amount of central expansion by stretching 130 degrees instead of 90 degrees to fill the 180 degree-long prime meridian, we obtain:

which, although it is still not perfect, is very nice for a conformal projection.

This projection suggests, however, that if one wishes to be completely unabashed about showing Europe on a larger scale than the rest of the map, one could construct a map like the following:

but Europe is not enlarged by all that much even on this map.

Also, the availability of a little extra space on the transverse aspect of August's projection shown above suggests the benefit of using only a very modest amount of central expansion (expanding 150 degrees to cover the central meridian) as shown here:

Here's an example of using another transformation, the compression used to produce the Lagrange conformal or the conformal conic, to attempt to reduce distortion, but it doesn't seem to have been successful:

As an example of the possible pitfalls one might encounter when working with complex numbers,

basing a projection on the equation 5z-z^5 instead of on 3z-z^3 does produce cusps of the right shape at the poles, but unfortunately it also produces two extra cusps on the Equator.

Since complex-valued differentiable functions produce conformal mappings, their sums remain such functions. If one takes two August's projections rotated by 90 degrees - with the world within them rotated as well - it should be possible to position the cusps so as to obtain a conformal projection with fourfold symmetry:

however, it is surprisingly not conformal at the South Pole, despite the fact that both component projections have sharp corners at that point.

Doing the same with two Lagrange's projections obtains the following:

which is interesting because the Northern hemisphere is given sharp corners; this allows one to avoid use of elliptic integrals to obtain the opposite distortion of the one provided by 5z-z^5, halving angles at four points instead of doubling them.

Also, note that the scale distortion is surprisingly moderate over most of the map for a conformal projection. However, Antarctica is shrunk almost to invisibility on this map. The fact that the extremes of scale distortion are manifest as shrinkage instead of enlargement also makes this an unusual conformal projection.

Attempting to sum together two August's projections, but with the world tilted forwards by 90 degrees in one, to get a projection with the two halves joined along a line 90 degrees in length instead of 180 degrees in length, to get less distortion by being closer to the case of two independent stereographic hemispheres, instead of doing so by the technique shown above of mapping Stereographic projections of different sizes on each other, obtains the following:

which is marred by the emergence of some unexpected cusps.

Incidentally, since most normal mathematical functions, when applied to complex numbers, produce conformal mappings, a projection called the GS50 projection was devised which started from a Stereographic projection, and then transformed it using a high-order polynomial the terms of which were determined by a computer optimization process, to produce a projection which had very low scale error for the continental United States, and for Alaska, and for Hawaii, while having higher error elsewhere, on one single continuous map.

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