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At the bottom of the article, it refers to a new map projection that's supposedly more accurate: > A new world map published in 2021 by physicists J. Richard G

by mcv 3y ago

At the bottom of the article, it refers to a new map projection that's supposedly more accurate: > A new world map published in 2021 by physicists J. Richard Gott, David Goldberg and Bob Vanderbei minimised such distortions. Interested, I read the article and try to understand what makes their Gott-Wagener projection more accurate than others. Apparently this is determined by calculating 6 types of distortions, and then summing the squared error of those distortions, but only after normalising on the Equirectangular projection. And I'm wondering: why that normalisation step? And why Equirectangular? Does that not introduce its own distortions? In particular, Mercator scores very poorly because in the category where it scores worst, Equirectangular scores pretty good, creating a big difference that gets emphasised by squaring it. So I recalculate the summed squares without the normalisation, and I get very different results: Normalised: not normalised: Equirectangular: 6 0.09959 Mercator: 8.308 0.46488 Lagrange: 6.556 0.32536 Briesemeister: 6.193 0.49412 Eckert IV : 5.8474 0.408 Winkel-Tripel: 4.563 0.18284 Gott-Wagener: 4.497 0.10969 So funny thing: Equirectangular is now suddenly the best, when it used to be average. Gott-Wagener is still better than Winkel-Tripel and only just after Equirectangular. Mercator is still bad, but Briesemeister is slightly worse. So I've got serious doubts about this normalisation procedure. I suspect you get very different results depending on which projection you use to normalise on.

jacobolus 3y ago
Gott & al. have a decent basic concept, but a highly questionable implementation of the idea. (At least, in the paper of theirs I looked at from a few years ago; this might be a slightly different one.) They optimize for least squares distance errors for all pairs of distances across the map, but they consider points which are at opposite sides of a map across a discontinuity (but nearby on the globe) to have extremely high error, and therefore their optimization routine radically tugs those points towards each-other. The result is really scrunched up edges. They'd get dramatically better results by either (a) ignoring very distant points altogether, (b) figuring out how to measure distance across a discontinuity, or (c) comparing distances on the map to distances on the globe if you go the "long way" around, i.e. following a similar path to the one implied to be shortest by the map.

by mcv 3y ago

jacobolus 3y ago

Gott & al. have a decent basic concept, but a highly questionable implementation of the idea. (At least, in the paper of theirs I looked at from a few years ago; this might be a slightly different one.) They optimize for least squares distance errors for all pairs of distances across the map, but they consider points which are at opposite sides of a map across a discontinuity (but nearby on the globe) to have extremely high error, and therefore their optimization routine radically tugs those points towards each-other. The result is really scrunched up edges. They'd get dramatically better results by either (a) ignoring very distant points altogether, (b) figuring out how to measure distance across a discontinuity, or (c) comparing distances on the map to distances on the globe if you go the "long way" around, i.e. following a similar path to the one implied to be shortest by the map.