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Welp. I'm confused.
A high min-scale Cylindrical-Equal-Area world-map with good shapes everywhere.
I propose a stacked combination of 3 Cylindrical Equal-Area world-maps.
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Cylindrical Equal-Area (CEA) maps have big advantages, and the disadvantage that each of them has really bad shapes at some latitudes. And the ones with good tropical shapes have relatively great NS compression and relatively small min-scale at high-lat.
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CEA has the advantage of being the 2nd simplest equal-area map. (Sinusoidal is the simplest, but people don't like its shape-distortions and small min-scale.) ...and the easiest one for which to draw the grid, because the grid consists of all straight-lines.
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Also, Cylindrical projections are able to give information, via the special ruler about which I've posted here, that other maps can't give.
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Different CEA versions differ in their aspect-ratio (ratio of EW/NS dimensions). Each version looks good at some latitudes.
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e.g. Behrmann shows the tropics with acceptable shapes, without too much flattening or loss of min-scale at high-lat.
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Tobler CEA is a square map, with equal NS and EW dimensions. (Its aspect-ratio is 1.0). Tobler has the advantage that, even up to Norway's arctic coast (North-Cape), the min-scale is no less than the scale along the equator. But the price for that is that tropical regions are hugely shape-distorted, with Aftrica and South-America looking even more implausiblly-skinny than in Peters. (...not that it could really get any more implausible than that.)
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And, in that way, Tobler also wastes lots of space to distort the tropics.
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Well, Tobler is good for high-lat, and Behrmann is good for low-lat. So then, why not use Tobler only where needed?
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Behrmann's min-scale is no less than the scale along the equator, up to lat 41.41 That's approximately the latitude of Mount Shasta and Barcelona, Spain.
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So why not map the entire Earth on Behrmann, and have, directly above it, on the page or map-sheet, a Tobler map showing the Earth from lat 41.41 to the North Pole?
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In fact, though this is less needed south of the equator, because the inhabited land doesn't reach as far toward the pole, you could do the same thing in the South. From minus 41.41 to the South Pole, have a map on which the min-scale at Cape-Horn is no less than the scale along the equator. That would be a CEA map whose aspect-ratio, if it mapped the whole Earth, would be about 1.8
That's 1.743 times less than that of Lambert's CEA map, and about 1.31 times Behrmann's aspect-ratio.
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This stack of 3 maps would have an overall aspect-ratio of 1.45. It would neatly fit on an 8.5X11 sheet of paper, with a little space left over at top and bottom in the 8.5-inch dimension.
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...and it would show every part of the Earth, from Cape-Horn to North-Cape, with local min-scale no less than the scale along the equator, and with acceptable shape.
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I doubt that there's any other non-interrupted equal-area world-map for which that can be said.
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I call that map "CEA-Stack".
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Michael Ossipoff
May 1st, 2020
Welp. I'm confused.
An example would have been useful, because I can't really visualize what the intended benefit would be without seeing one. I'm not sure what particular value is suggested by this complex construction over more common projections such as the Equal Earth projection ( https://en.wikipedia.org/wiki/Equal_Earth_projection ).
Oops! I should have checked sooner for replies. I didn't, because there were no repiles to previous post. I'm replying now, immediately upon finding the replies to my post.
First, I thank both of you for your replies. You make a good point--I didn't say enough in my post, and it would have been better if I'd posted the map, in addition to its description.
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You asked what CEA-Stack's advantge is. (CEA stands for Cylindrical Equal-Area). I admit that CEA-Stack is something of a special-purpose map. It's for when min-scale is really important. ...such as when it's necessary to make measurements of relatively small distances, or examine relatively small regions, or when it's desired for the map to be useful when viewed from a distance.
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Yes, sometimes, maybe usually, those aren't prime-considerations. But sometimes they are, and that's what CEA-Stack is for.
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With CEA-Stack, the whole region from Cape-Horn to North-Cape (Norway's arctic-coast) is shown with local min-scale nowhere less than the scale along the equator. That's relevant, because a map's equatorial-dimension is typically the fit-critical dimension. ...the dimension that determines how big a map will fit in the available display-space.
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The advantge described in the paragraph before this one is an advantage that probably can't be claimed for other equal-area world-maps.
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I should clarify what I mean by "local min-scale":
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In CEA (Cylindrical Equal-Area), of course, away from the equator, the EW scale begins to get much too large. To preserve equal-area, it's necessary to correspoindingly reduce the NW scale, for places farther from the equator.
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local min-scale at a point is the shortest scale at that point. On a map that isn't conformal (and no equal-area map is conformal), the scale, at any point, is different in every direction. Local min-scale refers to the shortest scale at a particular point.
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In CEA, the problematically-short scale is in the NS direction at high-lat.
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Of course in CEA, the local min-scale is the same at every point on a particular parallel.
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Low high-lat local min-scale is the bane of CEA (along with ridiculous tropical skinniness in some versions).
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The map consists of 3 sections:
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1) The middle section is a Behrmann map of the entire Earth. Behrmann gives adequate shapes for low and mid latitudes, and good scale. Only at latitude 41.41 does Behramnn begin to have local min-scale lower than the scale along the equator.
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So the top and bottom sections use more vertically-large CEA (CylindericL Equal-Area) versions, to achieve keep the local min-scale at least as large as the scale along the equator.
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2) The top section is directly above the middle-section. It shows from lat 41.41 to the North Pole, using Tobler CEA, whose aspect-ratio (ratio of overall EW dimension to overall NS dimension), when Tobler maps the entire Earth, is 1.0
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(Behrmann's aspect-rato is (3/4)pi, which is about 2.356)
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3.)The bottom section is direectly below the middle-section. It shows from minus 41.41 latitude to the South Pole. The CEA that's used for the bottom section is one which, if it mapped the entire Earth, would have an aspect-ratio of 1.8 ...in order to ensure that all the way down to Cape-Horn, the local min-scale is never less than the scale along the equator.
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As I said, I chose 41.41 as the latitude as the inner boundary of the regions shown by the upper & lower sections, because that's where Behrmann first starts having a local min-scale less than the scale along the equator.
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By the way, 41.41 is the approximate latitude of Mount Shasta (a mountain in Northern California), and Barcelona, Spain.
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But even ordinary CEA has advantages compared to other equal-area maps:
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1. It fits rectangular space better, allowing a bigger map in a given rectancular space. That means that every place on the Earthis shown bigger. That might not matter, but it might matter, depending on the application.
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2. With the use of the Position & Properties Ruler that I proposed and described, a Cylindrical map can give much more information than any other kind of map can. A Pseudocylindrical, such as Equal-Earth can give a little information, but not nearly as much as a Cylindrical.
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3. CEA is the 2nd simplest equal-area map. As I mentioned in my other post in this thread, Sinusoidal is the simplest, but people don't like its distorted shapes and small min-scale.
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You mentioned that Equal-Earth is simpler. Have you seen its formulas? More about it later in this post.
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You're quite right: I should have included the map in my post, to accompany my proposal and description of it. In fact, yes, I've never heard of someone proposing, offering or introducing a map without showing what it looks like.
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Here's why I didn't. I waited a long time to post about it, because I wanted to wait till I had time to make the map, so I could show it to accompany my introduction and description of it, But it was getting to be such a long time, that I decided to just introduce, propose and describe it now, and post the map later when I get the opportunity to make the map.
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.I certainly can't expect anyone to like a map that hasn't been shown. I'll post the map itself as soon as I get a chance to draw it. But of course I'll shorten the delay by just drawing the continents in, without drawing the national borders, rivers, lakes, mountains, etc.
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Yes, sometimes, often, or even more often than not, Mollweide would be preferable, because of its globe-realism...even though a CEA-Stack, or any ordinary CEA map, fits a rectangular-space better, thereby showing everything bigger, and even though Mollweide doesn't share CEA-Stack's advantage of showing everywhere from Cape-Horn to North-Cape with local min-scale at least equal to the scale along the equator.
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Likewise, often ordinary Behrmann would be perfecly adequate. Bigger than Mollweide, with (as I said) local min-scale at least equal to the scale along the equator up to lat 41.41, and a local min-scale equal to 2/3 of the scale along the equator at lat 60.
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CEA-Stack is just for when you want the best, biggest min-scale from Cape-Horn to North-Cape.
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Motivation:
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I know I said this, but let me repeat it: Behrmann is good because it looks good in the tropics, and even up at lat 60 its local min-scale is still 2/3 of the scale along the equator. Even up at North-Cape, its locak min-scale is still 43% of the scale along the equator. ...not as good as CEA-Stack, but still pretty good.
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And Tobler CEA has the advantage of local min-scale at least equal to the equatorial-scale all the way up to North-Cape, and good shapes all the way up there. But of course it has awful shapes in the tropics, and wastes a lot of space distorting the tropics.
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So why not just use Tober where needed? ...at high lat. That's what CEA-Stack does. CEA-Stack combines the best of CEA.
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Equal-Earth Projection:
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I wouldn't say that Equal-Earth is simpler than CEA-Stack, given the great simplicity of CEA-Stack, and the fact that there don't seem to be any articles on the Initernet telling how Equal-Earth's formulas were arrived-at.
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Well they do tell that the outer-meridian curves were chosen by experiment. ...looking at various curves, by using software that would draw the various curves for examination. When curves were found that seemed to result in overall best-looking appearance (or appearance most like Robinson?), then the curve-formulas that achieved that desired result were written-down.
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But what the net-aricles don't say is: How was the general form of the equations chosen?
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Robinson's Projection was designed by just looking at various combinations of outer-meridian-curves and parallel-spacings, to find which looked best. Then, the parallel-spacings, and the lengths of some of the parallels were recorded in a table. Those parallel-positions and parallel-widths were all recorded in tables, and those tables define Robinson.
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The situation with Equal-Earth is different, because, when you select an outer-meridian curve to look at, there's only one set of parallel spacings that will give equal-area for that outer-meridian curve. So the job is simpler than it was for Robinson, because one only has to find the outer-meridian curve that makes everything look best.
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As for the form of the outer-meridian curve-formula, the formula whose parameters were varied to make the various curves to try out: I don't know how they chose it, but it would need to be such that varying its parameters could give all the kinds of general shapes that they wanted to look at.
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But, beyond that, it could additionally be chosen so that it's easy to antidifferentiate, and so that its antiderivative, when written, is easy for the computer to evaluate. Maybe they started with possible formulas for the parallel-spacings, choosing a formula that is easily evaluated, and whose derivative is fairly simple.
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But no, that matter is not as easy to explain as the formula for CEA, and its derivation. ...since the websites don't say anything about how the general form of the curve-formulas were chosen. (I was just guessing, above.)
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The fomula for CEA is about as simple and brief as it gets. And the way by which that formula is gotten (its "derivation") is easily demonstrated to anyone who is willing to listen to the explanation. (...but this post is already too long.)
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But I'd be glad to post the derivation of CEA's simple and brief formula later, in a subsequent posting. As I said, it will make sense to you if you're willing to listen to it. Unlike the formulas for Equal-Earth, no calculus is needed for the derivation of CEA's formulas.
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Sorry that this reply is late. I'll post an image of the CEA-Stack map as soon as I draw it. I should likely have time to do so in the coming few days. As I said, I'll make the job briefer by just drawing the outlines of the continents.
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Michael Ossipoff
May 4th, 2020
0425 UTC
https://www.researchgate.net/publica...map_projection goes into pretty good detail about the rationale for Equal Earth.
There are a huge number of special-purpose projections out there. Finding one that meets the intended purpose of the projection while still being aesthetically pleasing is a very difficult problem.
Thanks for the link.
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The article certainly shows the motivation for the formulas; and it shows that there's a good reason why they're as they are; and tells at least a general explanation of how they were gotten. I mean, the article tells exactly how they were gotten, but verifying each formula-change in each step of the very long derivation would be a task that few of us would be inclined to take on.
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...and, as I said, it requires calculus.
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In contradistinction, the equal-area Pseudocylindricals whose outer meridians are straight-lines, circles or ellipses can be derived, and their derivation complerely explained, without the use of calculus. As I said, such maps can be explained to, and that explanation will make sense to, anyone who has the time and patience to hear the explanation, and is interested enough in the derivation to listen to it.
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And Cylindrical Equal-Area maps have a derivation that's much briefer to explain than any Pseudocylindrical Equal-Area map. ...and that includes CEA-Stack, which is just a stack of 3 CEA maps.
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I like things that I can explain to everyone. I reject map-projections whose formula-derivation or construction-derivation I can't explain to everyone.
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Speaking of Cylindricals and Pseudocylindricals, I should define those term:
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A Cylindrical projection is one whose meridians are straight-lines that are equally-spaced and parallel to eachother, and whose parallels are straight-lines that are parallel to eachother and perpendicular to the meridians.
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A Pseudocylindrical projection is one whose parallels are straight-lines that are parallel to eachother, with the scale along each particular parallel uniform all along that parallel. ...and whose meridians aren't all straight-lines. The central meridian is a straight-line that is perpendicular to the parallels.
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(I should add that some people leave-out, from the definition of a Pseudocylindrical, the part about the meridians not all being straight-lines. So, those people are saying that a Cylindrical is a Pseudocylindrical. To me that's nonsense. )
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Note that the parallels of Cylindricals and Pseudocylindricals needn't be equally-spaced, and that, on a Pseudocylindrical, the scales along different parallels needn't be the same as eachother.
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The Sinusoidal, the simplest equal-area world-map, is the Pseudocylindrical whose parallels are equally-spaced, with all of the parallels having the same scale along them.
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But, as I said, the Sinusoidal has drawbacks that people don't like. Because I value good min-scale, I wouldn't choose Sinusoidal either.
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You've seen the Sinusoidal. It's the roughly diamond-shaped world-map that's often used in a logo or a sign-display, but not often on mapsheets or in books.
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Yes, often a special-purpose map has an aesthetic drawback. That drawback might be necessary to achieve that special-purpose.
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CEA-Stack's aesthetic drawback is the stacking, which complicates the map's appearance, as compared to a single world map.
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But, if you want an equal-area world-map with the world between the latitudes of Cape-Horn and North-Cape to be shown with local min-scale not less than the scale along the equator, then CEA-Stack achieves that. ...and any aesthetic-disadvantage of the stacking is justified if that goal is desired.
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But I hasten to agree that often or usually, that high-min-scale isn't necessary, and so ordinary Behrmann, or the equal-area Pseudocylindricals are adequate.
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In particular, Mollweide, comared to the other Pseudocyilindricals and the Cylindricals, has a globe-realism that's unmatched. Mollweide's point-pole, with its point-converging meridians, gives a topological-realism that the line-pole Pseudocylindricals and the Cylindricals don't have.
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Though I like CEA-Stack because I value its advantage, and would offen want it, I'd probably choose Mollweide for most purposes, when detailed small-region measurements or examinations aren't needed, and where distant-viewing of small details isn't expected to be likely necessary.
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...but sometimes Behrmann, if map-size and min-scale are important enough to justify it over Mollweide, but if the min-scale best-ness of CEA-Stack isn't needed.
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Michael Ossipoff
May 3rd, 2020
2343 UTC
Originally Posted by waldronate
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