CEA best of both worlds

[RogerOwens]

The trouble with CEA is that you have to choose between ridiculously-shaped tropics and NS-squashed high-latitudes. …or do you?
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Behrmann has acceptable tropical shapes, and the min-scale isn’t too bad up to lat 60, though the shapes are already definitely wrong there. But those shapes don’t jump out at you like the tropical shapes.
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Anyway, Balthesart or Square Tobler CEA keeps the min-scale acceptable all the way to Europe’s arctic coast.
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So why not have a Behrmann map of the entire world, and have, directly above it, a Balthesart or Square Tobler CEA map of only some of the higher latitudes. I don’t know—maybe from 50 on up?
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That Balthesart or Tobler part would be a relatively thin strip directly above the Behrmann map, barely separated from it.
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Behrman’s NS dimention isn’t much in comparison to its EW dimension. And, given all CEA maps’ NS squashing at high latitudes, that Balthesart or Tobler strip at the top wouldn’t be very wide in the NS direction, since it’s only mapping from (say) lat 50 to the pole, and that region is somewhat squashed in any CEA.
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If you want to compare the area of a high-lat place to a lower-lat place, the Behrmann map is of the entire Earth. If you want a closer examination at high lat, then the Balthesart or Tobler strip shows that region with better min-scale.
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And all this without making the overall NS dimension too large in comparison to the EW dimension.
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Michael Ossipoff
July 26th
Week 31, Friday
0450 UTC

[RogerOwens]

Because Tobler CEA looks so awful in the tropics, but shows the far-north better than other CEA maps, then it’s natural to use Tobler only for the region where it’s needed, and not where its shapes are so blatantly bad (and would waste a lot of NS space).
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Hence my proposal, CEA-Stack. I was going to call it CEA-Combo, but “combo” is too vague, and could refer to a graft. “Stack” implies one map piled on top of another, reminiscent of an entry in a pancake-menu.
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Aside from the advantages of cylindrical projections in general, CEA is the 2nd easiest-to-explain equal-area world-map (after Sinusoidal, which suffers from unpopular appearance and low min-scale). And CEA-Stack has high min-scale for an equal-area map.
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I doubt that any other equal-area world map matches CEA-Stack’s min-scale up to the latitude of the northernmost arctic coast of Continental-Europe.
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How I rate equal-area maps:
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By “min-mcale”, I refer to a map’s smallest scale anywhere on the map, up to the latitude of the northernmost arctic coast of Continental-Europe.
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By “point-min-scale”, I refer to the smallest scale at a particular point (the scales at that point being different in different directions).
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So here are the ways that I’d rate equal-area maps:
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1. min-scale, as defined above.
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2. the mean point-min-scale over the whole map.
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3. the square-root of the map’s area, divided by the Earth's area (or, for CEA-Stack, the combined area of the 2 component-maps, divided by the Earth's area + the Earth's area above lat 50)
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I’d express all of those in terms of a reference-scale.
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What I mean by “reference-scale”:
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1. If the size of the map that can fit in a particular display-space is limited by the map’s largest EW distance, then the reference-scale is the average-scale along the map’s largest EW difference.
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2. If the size of the map that can fit in a particular display-space is limited by the map’s largest NS distance, then the reference-scale is the average scale along the map’s largest NS distance.
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3. If neither of the above two conditions obtains, then the reference scale is the square-root of (the area of the map’s circumscribing-rectangle divided by the actual area of the Earth-surface shown on the map).

(Of course, for CEA-Stack, the amount of Earth-area shown is more than the Earth's area, because the part of the Earth above lat 50 is shown twice.)
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Yes, it would be neater and maybe more aesthetically-appealing, to just have one reference-scale definition, for all conditions. But I’m more interested in practicality than aesthetics, for the purpose of defining reference-scale.
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Of the 3 ratings that I listed, I prefer min-scale. But mean point-min-scale is arguably of interest too. …the average scale-disadvantage instead of the worst one. Square-root of area might be a useful comparison between maps whose min-scale is at, near, or above unity.
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It seems to me that, among equal-area maps that don’t outrageously distort the topics, CEA-Stack is probably the winner by the ratings that I’ve described.
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Michael Ossipoff
July 28th
31 Su
1918 UTC

[Atarimaster]

Just to make sure that I’m getting you right…
Your map would look like this?

balthasart-behrmann.jpg (104.08 KiB) Viewed 1699 times

(That’s Behrmann combined with Balthasart)

[RogerOwens]

Hi Tobias--

Yes, but both maps would have the same width. And I've now decided that the north-band would be Tobler CEA, for the following reason:
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Of course Lambert CEA is very unrealistically flattened in the Arctic. Greenland is as flat as a pancake.
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But, though Tobler CEA is noticeably flattened some, that flattening at the northernmost arctic coast of Continental Europe is just enough to simulate the natural forshortening that would be seen when observing a globe in equatorial-aspect from a distance.
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Therefore, Tobler CEA's shapes are globe-realistic at Europe’s north-coast.
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So Tobler CEA is what I’d use for the north-band of CEA-Stack. Behrmann for the whole Earth, with an upper band of Tobler CEA from lat 50 to the pole.
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Michael Ossipoff
July 29th
32 M
1730 UTC

[daan]

It’s reasonable to represent regions of the earth twice in a single display. That’s a strong departure from the constraint of expressing the entire earth exactly once, and so I wouldn’t really think of it as comparable. Instead, it should be compared to, for example, the common practice of repeating the poles in polar azimuthal projections in the blank corners of a pseudocylindric or some other lenticular projection. I’d much rather use polar azimuthals for representing poles, and doing so still serves the purpose of “optimizing” usage of the rectangular space, if that’s a goal.

— daan

[RogerOwens]

Yes, as much as I like CEA, cylindricals aren't any good at all for regions near the poles, and those separate polar azimuthal maps (...which I realize could be equal-area of course) would be an addition that would better show the poles.

I guess, with a cylindrical, it's either add polar azimuthals, or else write-off the polar regions.

I guess point-pole pseudocylindricals like Mollweide or Quartic aren't too bad near the poles. ...another advantage of Mollweide, in addition to its in-general better globe-realism, compared to CEA.

Michael Ossipoff
August 2nd
32 F
2133 UTC

[RogerOwens]

But yes, corner room for polar azimuthals is another advantage of pseudocylindricals.

And I see what you mean about CEA-Stack having something in common with the use of those added polar-azimuthals--in both instances, a separate map is provided for a region otherwise not well-shown.

Michael Ossipoff
August 2nd
32 F
2140 UTC

[RogerOwens]

I should correct something that I said:
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When I defined my proposed equal-area map-ratings, I defined them in terms of a reference-scale. So, when the display-space is such that the map’s NS dimension is the critical (map-size-limiting) dimension, and the map is CEA-Stack, I’d determine that map’s average NS scale based on the NS map-size, and the amount of Earth-distance it represents, considering the fact that some of that is shown twice.
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But the whole point for me, what I ask, is: “What’s the map’s smallest scale, for a map that can fit a given display-space (in which the map’s size is limited by its size in one dimension or the other).
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So I what I want isn’t really the map’s average scale in that dimension; it’s its size in that dimension.
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Of course ordinarily it doesn’t make any difference which way that reference is defined, because the scale is calculated in relation to Earth dimensions that are constant. But, with CEA-Stack, it makes a difference. And, then, it’s clearly the distance, not the scale, that’s of interest to me.
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So I’d define min-scale as:
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Min-scale is:
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The smallest scale on the map below the northernmost arctic coast of continental-Europe, divided by the map’s greatest measurement in the critical-dimension.,
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[end of min-scale definition]
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Where the critical dimension is EW, or when there’s no display-space limitation, I like the CEA-Stack that uses Behrmann for the lower (complete) map, and Tobler for the upper (partial) map.
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That’s how I define the main version of CEA-Stack.
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Unless I made an error, with the critical-dimension EW or NS, that main CEA-Stack has larger min-scale than plain Behrmann.
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When the display-space is such that the critical dimension (the map-size-limiting dimension) is NS, then it’s a whole completely-different situation, and I haven’t thoroughly examined it.
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Of course if NS is critical, and the map can be expanded EW as much as desired, then it’s just a matter of personal taste how far one expands it. I’d say that, for looks (but also for EW scale), it should be expanded at least to be Behrmann.
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But why not expand it EW all the way out to Lambert?
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Advantages:
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1. Better tropical-shapes
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2. Larger EW scale everywhere
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Disadvantage:
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1. Bad effect on high-lat shapes
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2. If the map has EW vertical, then the map’s height could become such as to make some parts of it not easily accessible, especially if it’s intended to be used from a desk-chair.
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Subject to disadvantage #2, I’d probably choose the EW expansion to Lambert, but I don’t advocate it, and I emphasize that it’s a matter of individual preference.
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That’s the easy part. The hard part is the matter of choosing (when NS is critical) 1) the ratio between the radii of the construction-circles for the upper & lower CEA maps; and 2) the latitude at which the upper map starts.
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I guess there’s a way of choosing those parameters to maximize min-scale, but I haven’t looked at the problem.
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But, when min-scale is thereby maximized, I don’t know how good the shapes would be.
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My first guess for what I’d like would be to just use, for the radii-ratio and starting-lat for the upper-map, those of the main CEA-Stack version that I defined above. But maybe it could be improved, with NS critical.
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So, with NS critical, (and lacking more information) I’d just start with that main CEA-Stack version, and maybe (or maybe not) expand it EW out to Lambert.
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Of course, when one chooses any kind of CEA, one is choosing better non-arctic min-scale, and writing-off a good portrayal of the polar-regions and overall globe-realism.
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For better polar portrayal and globe-realism, I’d choose Mollweide.
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Michael Ossipoff
August 3
32 Sa
2038 UTC

[RogerOwens]

Tobias--

Your version with Tobler having less EW width than Behrmann is better!

Just because the low-lat section is improved (better scale and shape) by EW expansion, there's no law that says the high-lat section has to be expanded EW too, spoiling its shapes.

Each section should have the EW dimension that suits it best.

Michael Ossipoff
August 4th
32 Su
1601 UTC

[RogerOwens]

And so, likewise, if the low-lat section is EW expanded out to Lambert, there'd be no need to spoil high-lat shapes by doing the same to the high-lat section.

I don't know what the ideal construction-circle radii ratio would be, or the ideal lowest lat in the high-lat section, to maximize min-scale when NS is the critical dimension; or how that optimization would affect shapes--but shapes would almost surely always be helped by the flexibility of not having to make both sections the same width.

Michael Ossipoff
August 4th
32 Su
1614 UTC