CEA best of both worlds

[RogerOwens]

It has occurred to me that, though adding the Tobler section gives more high-lat fairness, the increased NS space it takes up leaves less for the main map (Behrmann, in my main CEA-Stack version), where the NS dimension is critical.

(Of course when NS dimension isn't critical, then there'd be no reason to not add the upper-section (the Tobler-section in my main CEA-Stack version) ).
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Behrmann, as-is, isn’t really unfair to populated high-lat places.
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Conformal at lat 30
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NS scale equal to equatorial scale around lat 41
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NS/EW shape-proportion on map matches that of globe view due forshortening at lat 41
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NS scale around 86% of equatorial scale at lat 50
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NS/EW proportions, 86%, compared to forshortened view on a globe, at lat 50.
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NS scale 2/3 of equatorial scale at lat 60
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NS/EW proportions 2/3, compared to forshortened view on a globe at lat 60.
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NS scale about 44% of equatorial scale at the north arctic coast of continental Europe
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NS/EW shape proportions 44%, compared to forshortned view on a globe at north arctic European coast.
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That isn’t so bad for single CEA map that shows the tropics without bad shape-distortion.
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There might be use for a CEA-Stack map, including one that actually minimizes min-scale (as I defined it) up the northmost arctic coast of continental-Europe, but, for ordinary general-reference purposes, when the NS dimension is critical, I’d prefer plain Behrmann.
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For that Behrmann-&-Tobler CEA-Stack that I spoke of, with the upper-section starting at lat 50, with the NS dimension critical, I’d keep both sections’ widths the same, though there could be other combinations of construction-circle radii ratio, and starting-lat for the upper-section, for which I’d use different widths. And, if the lower-section were expanded EW to Lambert aspect-ratio, I’d probably not expand the upper-section with it.
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Michael Ossipoff
August 7th
33 W
2318 UTC

[RogerOwens]

I’ve determined the min-scale, as I define it, of ordinary CEA and of two CEA stack versions: 1) the main CEA-Stack proposal, with Behrmann mapping the entire Earth, and Tobler mapping from lat 50 to the pole); and 2) the version that maximizes min-scale when the NS dimension is critical.
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I determined min-scale consisting of the lowest scale (on the map section where it’s least) for any place up to Europe’s arctic-coast (scale evaluated as map-distance per radian on the Earth), divided by the largest map-distance in the critical dimension (NS, in this case).
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Ordinary Cylindrical-Equal-Area: about .1611
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Behrmann-Tobler Stack-CEA, with Tobler starting at lat 50: .25195
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Stack-CEA optimized for min-scale: .269
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That scale-optimized version starts the upper-section map at lat 41.25, and its ratio of construction-circle radii is about 2.333
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Its min-scale is about 7% greater than that of the Behrmann-Tobler lat 50 version. Its ratio of construction-circle radii is within about a percent of that of the Behrman-Tobler version.
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I only looked at one local-maximum, and maybe the min-scale function has more than one local maximum…and so maybe the one I was looking at wasn’t really the overall maximum. But I looked at latitudes from 30 to 50, and the only local maximum I found was the one in which the upper map-section starts at lat 41.25
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It can be determined what construction-circle radii ratio maximizes min-scale for any given bottom-lat for the upper-section. So it’s only a matter of maximizing min-scale in terms of one of those two parameters, given a known relation between them.
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So, when NS is the critical dimension, evidently Stack-CEA can give a lot more min-scale than does plain CEA.
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So, with NS critical, I’d use the scale-optimized Stack-CEA described above.
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I doubt that any other equal-area world-map can match its min-scale up to Europe’s arctic coast.

I should add that yes it might be a good idea to EW narrow the upper-section as Tobias (Atarimaster) suggested, because 1) Doing so could make the two map-sections equal-area-consistent with eachother (a desideratum that hadn't occurred to me); and 2) It might help upper-section shapes.

The optimization that I spoke of is only about the two parameters I named, which affect NS scale, which is ordinaily the low one in CEA. The choice of EW dimensions when NS is critical is another matter, in which the person has free choice.
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Michael Ossipoff
August 13th
34 Tu
2146 UTC

[RogerOwens]

Because the inhabited land where CEA has serious scale-shortening are in the far north, then Optimal CEA-Stack is optimized for the northern-hemisphere. But, for that reason, the scale (map-distance per radian on the Earth, divided by the map’s largest distance in the critical dimension (NS in this case) ), at Cape-Horn is only .20
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So it wasn’t really correct to say that Optimal CEA-Stack’s min-scale is .269 Its lowest scale is only .20 at Cape-Horn.
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That could remedied by having a high-lat map-section in the South, corresponding to the one in the North.
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Just as the parameters-combination for north of the equator was optimized to maximize the min-scale out to North-Cape, then so the parameters for south of the equator would then be optimized to maximize the min-scale out to Cape-Horn.
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Michael Ossipoff
August 16th
34 F
1955 UTC

[RogerOwens]

Of course how bad scale is when it’s at its worst is more noticeable than how good scale is when it’s already good. That’s why my Optimal CEA-Stack proposals are optimized for min-scale (in inhabited regions or in ones that otherwise have the most significant scale problem).
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But average-scale (av-scale) is of some interest to me too, and, when the high-scale regions are the ones of most interest, then best av-scale could become the thing to optimize for. In fact, under those conditions, plain CEA could be a better choice than any CEA-Stack, even for min-scale.
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So, for plain CEA and Optimal CEA-Stack, I checked the average NS scale (NS map-distance per radian on the Earth, averaged over the entire Earth, divided by the map’s NS overall-map-distance).
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Values:
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Plain CEA: .3927
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Optimal CEA-Stack: .30
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So, if av-scale is what is of interest, then plain CEA is better.
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Of course Optimal CEA-Stack is optimized for min-scale, not av-scale. Of course CEA-Stack could be optimized for av-scale, in (say) the lat-region between North-Cape and Cape-Horn…or between the northernmost mainland continental city and Cape-Horn.
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I’ve emphasized the situation in which NS is the critical dimensions and where therefore NS scale is what is limited by the available display-space, because that fit-situation is the more demanding one, because CEA’s low-scale problem is about NS scale; and also because, if the main desideratum were getting the best min-scale for a given display-space, then it would make sense to have the map’s smaller dimension (its NS dimension) fit to the display-space’s limiting-dimension.
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But of course sometimes a map is to be fit to a display-space that’s limited in the map’s EW direction, for the orientation that you want the map in. And sometimes the map’s size is limited, not by the display-space, but only by viewing-convenience or paper-availability.
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Obviousy those two situations would suggest other combinations of considerations for which CEA-Stack’s two parameters could be optimized, for the best combination of shapes, with min-scale or av-scale.
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Michael Ossipoff
August 18th
34 Su
0058 UTC

[RogerOwens]

I’ve just noticed that, with Optimal CEA-Stack, in the upper-section, at the lat where the upper section starts, the NS/EW scale-disproportion is about 1.76 That’s nearly as bad as Peters' worst (2.0) and a lot worse than Behrmann’s worst (4/3). The whole point of CEA-Stack was to avoid something like that.
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When I first suggested optimizing CEA-Stack for best min-scale, I said that I didn’t know what that would do to shapes. It turns out that the shapes at the bottom-part of the upper-section aren’t really so good.
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My original CEA-Stack, where the upper-sections starts at lat 50, suffered from the problem that the NS scale just below that latitude in the main (lower-section) map was less than its value, in the upper-section, at North-Cape. Mid-lat places should have at least as good NS-scale as North-Cape.
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So neither of those CEA-Stack versions really brings the improvement that I was looking for, and so I don’t really know of a CEA better than Behrmann.
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But it seems to me that Behrmann beats the non-CEA equal-area world-maps in regards to min and av scale.
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Michael Ossipoff
August 21st
35 W
2304 UTC