- Comparison of Eisenlohr to August.
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The first projection is Eisenlohr’s. The second is August’s epicycloidal. The two inventors argued heatedly and publicly about which was better, back in the 19th century when they published their work.
Eisenlohr’s projection fulfills Chebyshev’s requirement for least variation in scale. That means it is, provably, the best conformal projection having only one interruption along one complete meridian. Best, that is, by the min/max criterion. The ratio between the least and greatest scale factor across the entire map is as low as it can possibly be.
August’s projection is simpler to calculate. Interestingly, it has lower least squares error (lower average distortion by some measure of average) than Eisenlohr. The lowest scale factor on both maps is 1.0 and only increases from there, which means both maps are larger than any equal-area map would be if the equal-area map’s Tissot ellipses have the same area as the point on the conformal maps that has scale factor of 1.0. The conformal map which is smaller must have lower •average• distortion, since an equal-area map represents no areal inflation, and the one that is closer in area to an equal-area map is the one that is closer to equal-area. Clearly August’s is smaller and therefore the average scale error is less. The difference is not just subtle, either; the August is considerably smaller.
However, in the region of the poles, August has more distortion than anywhere on the Eisenlohr map. You can see this in the highest parallel. The area above it on the August is larger, and the closer you get to the pole, the greater the inflation. This means that ratio of the lowest distortion to the highest distortion is much greater on August than on Eisenlohr. Furthermore, August cannot prove that his projection has the lowest least-squares distortion; he can only say it’s lower than Eisenlohr’s. Meanwhile Eisenlohr can prove his has the lowest min/max distortion.
Which one has less distortion?
There is no answer to that question. It’s not a well-formed question. It depends on how you measure it. Now, conformal maps are comparatively easy, and these two projections are so similar that you’d think a question like that is a piece of cake to answer. It’s not. It’s not answerable. It’s even less answerable when the maps aren’t conformal, because then they are less constrained, so there are even more ways to measure distortion. That’s not to say you cannot do it; people regularly publish such comparisons, but they always involve arbitrary or tendentious choices.