A flawed "cartometry"

[RogerOwens]

I realize that there’s no always-applicable merit-order among the equal-area pseudocylindricals. If someone were more interested in the tropics, then Eckert IV, in comparison to Wagner IV, could be better for them because, at peripheral longitudes, it favors low latitudes more by not having as skinny Tissot-ellipses…by not having as low min-scale.
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…even though, because it’s more cylindrical and less Sinusoidal, Eckert IV has more mid-longitude shape-problem—something admittedly a bit unrealistic. But, just as we accept that for various other equal-area pseudocylindricals, that person interested more in the tropics has good reason to accept a skinnier Africa in return for better min-scale at low-lat peripheral longitude.
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But it’s usually a premise with world maps that we don’t know what part of the world someone (even ourselves) is going to want to look at, at some time. So there’s the matter of balance and compromise among regions.
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For that consideration, Wagner IV seems to win the comparison, with its outer meridians that don’t become horizontal, and without the min-scale getting really bad in the Arctic. …and less mid-longitude shape problem, due to being more Sinusoidal-like and less cylindrical-like.
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What about Eckert’s and Wagner’s sine-edged maps? They have even better arctic min-scale and maybe they could be expected to have less mid-longitude shape-problem too. But they take a good thing too far, as perceived by most people. As with Sinusoidal, though to a lesser-degree, people don’t like the medium-latitudes, the temperate latitudes, to be so inward-swept-looking. And it also results in lower min-scale at middle, temperate, latitudes. …too high a price to pay for further improvement of the Arctic.
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So that leaves Wagner IV as the winner among general-purpose equal-area pseudocylindricals where it’s important to avoid low min-scale as much as possible, for the purpose of accurate precise measurements and examination, or maybe for a small map in a book, or for viewing from a distance—as with a wall-map in a classroom.
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It’s notable that Eckert was a geographer (whereas Mollweide was a mathematician and astronomer), and his introduction of his series of line-pole pseudocylindricals likely had something to do with their practicality for close geographical examination and measurement.
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Where those things aren’t necessary, I like Mollweide’s globe-realism and greater topological-accuracy. My girlfriend, without any coaching from me, chose Apianus II over Robinson, and chose Mollweide over the flat-polar equal-area pseudocylindricals, because of its more plausible, more realistic, more expected, more Earth-like look.
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Michael Ossipoff

[daan]

Among the interrupted projections Capek lists a projection called BSAM. Yet the only projection bearing this name that I know of is an uninterrupted cylindric projection. So which projection is he referring to?

I’m not sure, offhand. My books are boxed up right now. As is fairly common among Russian projections named for the institutes that developed them, probably several exist, and calling one BSAM doesn’t disambiguate. I think I took my appellation from Maling. If you prod me about this again in a few months I should be able to cough up a more satisfactory answer.

My books are shelved.

Maling, 2nd Edition, p. 433:

5b BSAM projection. First described in 1937. Used for maps in the Bolshoi Sovietskii Atlas Mira (Great Soviet World Atlas). The version of Braun’s projection (5) with standard parallels in latitudes 30°N and S.

Maling lists no others by the name BSAM. However, this from Snyder, Flattering the Earth, 1st edition, p. 195:

Eckert VI is the basis for climate maps in European-prepared but U.S.-distributed Prentice-Hall world atlases of about 1960; it was a staple, however, for the 1937 Russian B.S.A.M., where in the interrupted and condensed form it almost equally shared the numerous world-distribution maps with the modified Gall stereographic projection. It was called the BSAM pseudocylindrical projection, although credited to Eckert, and was also the basis of the map of the combined Pacific and Indian oceans in the same atlas.

I believe that clears things up. — though I don’t know what the interruption scheme and compression are.

Best,
— daan

[Atarimaster]

I believe that clears things up.

It does, thank you!

[RogerOwens]

It had occurred to me that if a map had a shallower circle-segment than that of Eckert IV, then maybe it would combine the advantages of Eckert IV and Wagner IV. I've noticed that Hatano's shallow ellipse-segment achieves that. (Maybe a circle wouldn't do as well).

Hatano seems to combine the advantages of Eckert IV and Wagner IV, and beat both of them, in regards to peripheral-longitude shear at middle and high latitudes, without having significantly or noticeably more middle-longitude shape-problem.

Hatano's north-south asymmetry seems justified because fewer people object if the antarctic is more squashed.

Then I noticed, at the Geocart projections-list, Urmayev II, which seems to benefit from the longer pole-line and shallower east & west boundary-curves, by having less peripheral-longitude shear, but somehow without its central-meridian shapes looking significantly worse. ...suggesting that it's possible to push the map-shape compromise closer to cylindrical-ness without getting the fully objectionable shape-proportions that put everyone off from Cylindrical Equal-Area.

(Peripheral-longitude shear is of interest to me because it results in a lower min-scale at peripheral-longitudes.)

Michael Ossipoff

[Atarimaster]

Speaking of Eckert IV/Wagner IV…
Did you see the new Equal Earth projection?
– Short introduction at shadedrelief.com
– Full introduction available as PDF at researchgate.net

[RogerOwens]

Hi Tobias--

I ran across it, but I didn't understand the details of the motivation for its formula, other than I guess that it was to approximate a drawn curve, as with Robinson, and make equal-area, with relatively little computational time.

It seems to me that a circle or ellipse is more closely-related to a globe's circle, and that equal-area pseudocylindricals using circles and ellipses have been used since 1805, and so I don't understand the need for someting more easily computed.

Equal-Earth's pole line seems 60% of the equator-length, placing it between Wagner IV and Hatano, in that regard.

But isn't Hatano still the winner, compared to those mentioned here, when it comes to the perpendicularity of parallels and meridians near the outer-meridian, and probably when it comes to the min scale at points near there?

If any of this isn't correct, that's because I wanted to reply right away (late hour here) before checking more thoroughly.

More tomorrow.

[Atarimaster]

I ran across it, but I didn't understand the details of the motivation for its formula

Well, as far as I get it…

There are numerous equal-area projections, and most of them show better continental shapes than Gall-Peters.
Yet, when the Boston Public Schools recently announced that they’re going to use Gall-Peters for their world maps, a lot of articles and comments on the Web praised the Gall-Peters, claiming that it’s the only "accurate" world map and that all we’ve ever seen before is the Mercator projection and blah blah blah… very much like the articles that praised the AuthaGraph map about a year earlier. Many articles include the notion that cartographers have struggled for centuries with the problem that "all" world maps inflate areas but never succeeded in finding a better projection.

Observation No. 1: Most People just don’t know that there are a LOT of equal-area projections or even that there are other equal-area projections besides the Gall-Peters at all.


Moreover, Mr. Savric and Mr. Jenny (two of the creators) also authored a paper called "User preferences for world map projections" in 2015.
Personally, I don’t agree with all the conclusion of this study – or rather, I don’t quite understand them, which I’ve mentioned in a posting some time ago –, but one of the results was:

Observation No. 2: A lot of people like the Robinson projection.

And I directly continue with
Observation No. 3: There was actually no need for a new formula.

Savric and Jenny are well aware of the customizable Hufnagel projection (because they wrote a paper about it). Using the Hufnagel interactive tool, they just could’ve dragged some sliders and finally coming up with
A = 0.189 / B = 0 / Ψmax = 49° / Length Ratio = 2.05
which is really close to the Equal Earth projection:

equalearth-vs-hufnagel-small.png (92.39 KiB) Viewed 1720 times

The red lines show the Hufnagel projection with the parameters given above (Note: generated in Geocart which uses the length ratio the other way round, I’ve used 0.4875 here), black graticule lines and grey continents show the Equal Earth projection.
There still are some minor differences to the final Equal Earth projection (you can see them better in this high resolution image, 4125 x 2031, approx. 1.6 MB), but keep in mind that

a) I just stopped trying to find an Equal Earth approximation here, it’s entirely possible that you can get even closer using the Hufnagel projection; and
b) I was trying to approximate the final result which of course didn’t exist when the creators started their work. And my Hufnagel configuration surely meets all the goals that are stated in the introduction of the Equal Earth projection, except for maybe the "easy-to-implement" part (that's something I can’t evaluate).

But, what is likely to attract more attention?
"We’ve found a pleasing configuration for the almost 30-year-old Hufnagel projection" or
"Hey, here’s a new equal-area projection called Equal Earth"?

So, that’s it.
The Earth Earth projection is not about optimizing distortion patterns – actually, the creators clearly state that McBryde-Thomas Flat-Pole Sine No.2 and Eckert IV have lower scale distortion and angular deformation values – it’s about raising attention for a visually pleasing equal-area projection that might be accepted by map readers easily because of its resemblance to the Robinson projection.
Well, that’s my interpretation anyway.

Thus, I think that the Equal Earth projection is a great idea, although personally, I still prefer other equal-area projections.



P.S.:
On a funny side note…
One of the articles praising Gall-Peters and condemning Mercator because it’s the only map that people know – is actually decorated with a photograph of a man carrying a world map that uses Robinson projection, along with two orthographic aspects, two polar equal-area azimuthal projections and two maps using the (equal-area!) Goode homolosine projection.
I bet the author didn’t even notice.



Note: In the original version of this posting, I wrote "intention" instead of "attention" once. I corrected this several hours later.

[RogerOwens]

Yes, its true that Equal-Earth looks a lot like Robinson. I can hardly tell them apart until I compare Greenland's size to the part of South-America that's north of the Equator.

Maybe Eckert IV beats Equal-Earth and Wagner IV with regard to mean min-scale, but not near the top of the outer meridians. ...not where it's at its at its worst among places other than the pole itself.

For an equal-area world map needed for precise observations and measurements about places close together, or for relatively-distant viewing, I'd choose Hatano, or maybe Urmayev II.

Maybe in many practical regards, Cylindrical Equal-Area (CEA)might be the best equal-area cylindroid map*, but everyone seems to agree that it looks unacceptably bad.

*(At least intuitively, it seems as if, if you start with a CEA map, looking at one mid or high latitude, and if you shorten the pole-line, curving the meridians inward (while adjusting the parallel-spacings to keep equal-area) that will help central-meridian shapes and central-meridian min-scale. ...while making min scale worse at the outer meridians. ...and thereby worsening the map's min(min scale) for regions below some specified high latitude such as +/- 60 or 70. That's maybe more obvious for equidistant-parallel maps, but intuition suggests it for equal-area pseudocylindricals too.)

No doubt Huffnagel is great for PhD mathematicians. I doubt that it can be of much value or interest to the rest of us, as a published map. Only as a way of investigating possibilities, ways that a chosen kind of map could look.

Michael Ossipoff

[daan]

Well, as far as I get it…

What a fine posting, Tobias!

Hufnagel is, in truth, harder to implement. It is not a big deal in the big scheme of projections, particularly after the paper by Jenny, Šavrič, and yours truly, but if you really just want something that looks like Equal Earth, then its own dedicated formulæ are easy.

— daan

[Atarimaster]

What a fine posting, Tobias!
Hufnagel is, in truth, harder to implement.

Thank you, for clearing that up and for the compliment!

… although I just had to correct a stupid mistake.