How do publishers actually interpolate for Robinson?

[RogerOwens]

Because I used only one parabola, for the entire half-meridian from pole to equator, on NGS Winkel and Robinson, for measuring the area of their map-quadrants, maybe this would be a better wording for what I found, regarding arctic magnification and space-efficiency:

For Eckert III:

arctic magnification: 2.58

space-efficiency: .893

Some likely lower bounds on arctic magnification:

NGS Winkel: 1.9

Robinson: 1.75

Some likely upper bounds on space-efficiency:

NGS Winkel: .82

Robinson: .869

Michael Ossipoff

[daan]

• Eckert III magnifies the Arctic by a factor of 2.58
  NGS Winkel magnifies the arctic by a factor of about 1.9
  Robinson magnifies the arctic by a factor of about 1.75

Actual values are

• 2.58437
  2.07816
  1.7574

1. NGS Winkel would be a very poor choice if accurate portrayal of the arctic’s share of the Earth’s surface-area is important.

It’s a matter of balance of concerns. Obviously an equal-area projection would be required for accurate portrayal.

it doesn’t really matter that most atlases make a poor choice

Poor by your personal criteria, unrelated to the goals of the publisher.

But aesthetics is the main claimed justification of Robinson (“orthophanic”, “right-appearing”). Does the Robinson Mystery-Map look good or right? Sure, except where it doesn’t. But we can just ignore those regions, right?

It’s quite clear Robinson did not share your goals or æsthetics, so all this harping on what doesn’t look right to you is seriously pointless. Whatever projection you might favor has gross faults to criticize as well, so can we please be done with these silly digs? Robinson didn’t pretend his projection looked right to everyone so there’s no point in interpreting “orthophanic” as if it’s intended to apply to everyone. It’s just one of a bazillion projections, one that happened to have received a larger share of attention than a lot of others, and largely for arbitrary reasons. I’m not going to defend its use, but neither am I going to allow unbalanced criticism to go unanswered.

What if a map magnifies some regions?… So what? It means that the map has better resolution in those magnified regions, and that can only be a good thing.

No, it’s not a good thing to represent some areas disproportionately. How is that a defensible proclamation? It’s merely a fault to put up with in order to gain other benefits. You can pretend to make lemonade out of that lemon by exclaiming that it gives more room for examination, but unless the audience specifically wants that benefit for that region, then no, it’s just distortion and therefore unfortunate. We don’t make globes with some parts squished and other parts magnified, for rather obvious reasons.

Winkel chose the standard parallel that yields the correct overall area for the map.

With respect to or in reference to what?

Its nominal scale, where the points of conformality take on unit scale.

So in NGS Winkel there are some European countries with good shapes and directions. …which, in NGS Winkel, certainly can’t be said for the U.S., or continents other than Europe. …or ¾ of the Earth’s surface.

Despite your insensitivity to distortion on some projections, all projections have this problem.

I don’t think disagreeing preferences can be called prejudice

That depends upon whether the person recognizes that “his” preference represents an æsthetic choice from amongst many competing, inaccurate representations, or whether instead the person believes his preferred representation is the accurate one. In the latter situation, the preference is a prejudice. In the case of the Mercator preference, it’s fairly clear the preference was a prejudice in large swaths of the populace. In your case, I would suggest some prejudice mixed with a lot of preference, with prejudice arising out of insensitivity to distortions that trained others recognize readily when they see them. That insensitivity extends well beyond the natural perceptual limits of about 5% error.

I don’t agree that people need to be educated to different preferences.

It’s not a matter of educating people to different “preferences”. It’s a matter weaning them off of any particular false belief about how things look and how projections work. If people’s exposure to world maps is never dominated by any particular projection or presentation, false beliefs about how things look can’t ever gel. Instead they’ll (at least subliminally) recognize that they all look different because none of them are right and none of them can be right. Possibly some “average” view of all them will impose itself into their consciousness, and that average view should be closer to a globe’s view than any single narrow class of projections would afford them.

For example, if maps other than Robinson and NGS Winkel are convincing people that the Arctic is a lot larger than it is, with serious adverse environmental consequences, and if that harm can be attributed to the maps, then you could say that preference for those maps is undesirable in terms of its results, and is a result of not considering the necessary information. But has it really been shown that that’s so?

No, it hasn’t, and that’s not my point. My point is that you cannot make credible claims by discounting concerns you’re not troubled by. My position is that you can use any “reasonable” projection more or less responsibly, where responsibly means you recognize the full gravity of what you have sacrificed in order to attain what you favored. I certainly did not mean to say that all world maps must show the arctic regions with better proportions; rather, I meant to say that some large fraction of a general audience would benefit by addressing concerns that you yourself discount. Because of that, it’s folly to make claims like “Eckert III is superior to Winkel tripel because blah blah blah”. The fact is, you give up some things that are important to some concerns no matter what projection you use, and just as importantly, whatever distortions are present in the projection will mislead all but the most highly trained audience. You cannot predict the consequences of that.

In my posting, I suggested a way to compare their magnitudes: I suggested that the shape distortion could be regarded as being in the amount needed to fix the areal-infidelity referred to. …But really here is the best way to compare them…

Neither this method nor the one you subsequently present are “uniquely correct”, and like map projections themselves, you give up something no matter what method you use. Many have been proposed.

But is an artic magnification of 2.58 “grotesque[ly]” larger than an arctic magnification of 1.9?

I never said “grotesquely larger”, and again, you have missed the point. What I did say was,

It’s about disproportion, and certainly people notice disproportion and even think it grotesque in some contexts.

And then, with respect to the arctic,

Eckert III gives the impression that there is a lot more arctic than there actually is

And with respect to Greenland,

…To me the relative inflation of Greenland on the Eckert III is huge, and even the inflation on the Bartholomew (Times) version of Winkel tripel is blatant compared to Winkel’s preferred version of his tripel projection.

And yet you keep harping on about the arctic as a whole when I specifically wrote Greenland:

Eckert III’s arctic magnification factor is 2.58 as opposed to NGS Winkel’s 1.9

Would you call that a “huge” difference in arctic magnification factors?

I was talking about Greenland, not the arctic as a whole.

More than Eckert III’s 2.58 magnification of the Arctic?

I was talking about Greenland, not the arctic as a whole.

The difference between 2.58 and 1.9 doesn’t mean much to me. Does it to you?

I was talking about Greenland, not the arctic as a whole.

If you need some graphical demonstration, here:

Winkel tripel (Winkel), Winkel tripel (Bartholomew), Robinson, Eckert III, “undistorted”

Greenland.jpg (37.5 KiB) Viewed 2466 times

If Eckert III’s representation of Greenland does not strike you as grotesquely disproportionate as well as obviously more severe than the others, well, perhaps we should conclude this part of the conversation. And by the way, by disproportionate, I do not mean absolute size, I mean change in proportion across the region. It’s just a mess.

But yes, Eckert III inflates the arctic a quarter more than Winkel tripel, and yes, that’s significant and obvious.

No, I can’t say if others agree. But it doesn’t take an expert cartographer to notice, chuckle at, or be put-off by, a real funny-looking wrong shape.

Yes, but I chuckle at distortions that you do not care about, so this is not a useful observation.

Well, how good are the sales-share percentages of NGS Winkel and Robinson wall-maps? You say they usually aren’t usually stocked in stores?

No, you said that. I have no idea, but rather than this,

Why aren’t they? Because no one wants them. …Ok, I was just trying to guess why nearly _all_ atlases use Robinson as their main world map. It seemed to me that the best explanation was that cartographers advise them to. Maybe it’s coincidence. Or maybe publishers got together and took a vote on what world map projection they should all adopt as a group.

…the answer is far more likely to reside in the fact that publishers rushed to follow National Geographic into the Robinson projection because of the fanfare National Geographic made out of their switch in 1988. Whereas, when National Geographic switched to Winkel tripel, they said nary a word about it. Meanwhile publishers had invested in Robinson and probably don’t see any benefit to retooling their cartography to the Winkel tripel. Some of them probably aren’t even aware that National Geographic switched. National Geographic sells millions of maps. If they were losing sales because of their projection, they’d be switching. The simple fact is that the buying public is nowhere near sensitive enough to differences between Robinson and Winkel tripel to affect sales.

…given that those preferences are mere prejudices, their relevance to a responsible choice of map projection is questionable.

Anyone who wants to claim that a map projection choice is irresponsible needs to specify in what way it’s irresponsible, and then demonstrate the correctness of that claim.

You’ve missed the point again. I was not saying that any specific projection is “irresponsible” (although you could include those that really are in my meaning as well); I was saying that pandering to people’s prejudices for any specific projection is a questionable way to choose a projection, whereas a responsible way is to disabuse people of their prejudices. That is, to educate them.

I think that it’s widely-understood that only a globe can accurately show the round Earth, and that flat maps always distort. Otherwise, would globes sell?

It’s not clear how widely it’s understood. I’d like to know.

Like any projection, it’s horrible as a world map

Isn’t that an extreme denunciation?

No.

Sure all flat world maps distort, but when that distortion is specified and known, how bad is it really?

Really bad. The general public has little sense for the distortions.

As a cartographer, you well know of the many reasons why flat world maps are useful, valuable, and even necessary.

I’m not arguing against world maps. I’m arguing against rubbing the public’s nose into any restrictive set of projections. I’m also arguing against personal biases being used as justifications for projection X and against projection Y when there are objective benefits to both and severe problems with both.

Best,
— daan

[RogerOwens]

Just to clarify, for anyone looking at these posts, what two projections we’re discussing and comparing, here they are:

Eckert III






NGS Winkel:






Forgive the size-difference. I couldn’t find a big image of NGS Winkel.


Let me just briefly say, in one place, the main things that I’m saying:

[beginning of brief summary and question]

1. Regarding Robinson, it would be nice to disclose a map’s projection. It would be nice for a “map” to be a specified mapping of lat,lon to X,Y.

2. Regarding NGS Winkel, it would be nice for representations to look like what they’re supposed to represent. In that regard, NGS Winkel, with its big disparity of N-W vs E-W scale, over ¾ of the Earth, is the worst, except for Peters and some of its close relatives (other north-south expanded CEA maps).

You speak of “balance”, and say that other maps are just as bad in their own way. In other words, you’re saying that NGS Winkel has significant advantages compared to other maps, and to Eckert III in particular . Ok, then…what advangages? . In the paragraph before this one, I mention what’s wrong with Winkel. Ok then, what is it trading for, in the trade-off. What is it getting in return for its unusually extreme and widespread shape-distortion, by which it badly, unusually, messes-up ¾ of the Earth?

The only thing you’ve named so far is that NGS Winkel only magnifies the Arctic by a factor of 2.08, as compared to Eckert III’s 2.58 Is that really the best thing you can say for NGS Winkel vs Eckert III?

I’d say that what _can_ be said for NGS Winkel is that it shows Antarctica better than many other maps, including Eckert III, do. But Antarctica is only 1/36 of the Earth’s surface. So NGS Winkel messes-up ¾ of the Earth in order to improve 1/36 of the Earth? …a 1/36 of the Earth that people refer to less often in general-purpose map-use, because there’s no country there and no one lives there, and very few visit there?

That concludes the question that I ask. By what advantage or accomplishment, in comparison to Eckert III, can NGS’s Winkel’s unusually widespread and extreme shape-distortion be justified??

[conclusion of brief summary and question]


Just to show these arctic magnification figures again:

Eckert III magnifies the Arctic by a factor of 2.58
NGS Winkel magnifies the arctic by a factor of about 1.9
Robinson magnifies the arctic by a factor of about 1.75

Actual values are
2.58437
2.07816
1.7574

Thanks for the precise values. I’d said that my figures for Winkel and Robinson were probably low.

1. NGS Winkel would be a very poor choice if accurate portrayal of the Arctic’s share of the Earth’s surface-area is important.

It’s a matter of balance of concerns.

Yes, and it would be worthwhile discussing exactly what is being balanced, what is being traded-for in the tradeoff—for NGS Winkel vs Eckert III. For comments on that, see above.

So, forgive me if I question the merit of NGS Winkel’s “balance” and tradeoff. I suggest that, for these reasons, NGS Winkel is objectively not as good as Eckert III, Times Winkel, or Oxford Winkel.

Obviously an equal-area projection would be required for accurate portrayal.

Yes, that’s what I meant. For anyone studying the Arctic, in regards to the things you mentioned, when area-proportionality matters, NGS Winkel wouldn’t do. Equal-area would be needed. But otherwise, for general purposes, 2.58 arctic magnification isn’t importantly more than 2.08.

Later, you call it a significant difference. But “significant” must be measured in comparison to other considerations, and of course ultimately each must make that evaluation for hirself.

You speak of a factor of 1.24 (between the two projections’ arctic magnifications) as being significant. Yes, if one projection didn’t magnify the Arctic at all, and the other magnified it by a factor of 1.24, that might be more significant. But when one is already magnifying it by a factor of 2.08, and the other only raises it to 2.58, how significant is that (even if it’s still more by the same factor of 1.24)—compared to the unusually widespread and extreme shape-distortion that I keep mentioning?

What if a map magnifies some regions?… So what? It means that the map has better resolution in those magnified regions, and that can only be a good thing.

No, it’s not a good thing to represent some areas disproportionately. How is that a defensible proclamation?

It certainly goes against current consensus among cartographers. That I admit. It goes against the prevailing assumptions. But all assumptions are subject to question. …even if questioners don’t make themselves popular when they ask if the Emperor really has any clothes. I ask you to disregard popular assumptions and preconceived notions.

Different magnifications in different regions would be a bad thing if one expects area proportionality, uniform magnification. But that’s an unrealistic expectation for a flat map of a round Earth. If you say that people expect the areas to have the proportions shown by the map, then you must also believe that people likewise expect that the Earth is flat, because the map is flat; and that the Earth is shaped like the map. People know that the Earth isn’t like the map in every regard
.
Additionally, different magnifications in different regions would be undesirable if the map is to be used for comparison of areas. For that purpose, obviously equal-area is needed. I often use an equal-area map for area comparisons. No, I don’t use Eckert III or NGS Winkel for that purpose.

For maps showing how much of the Earth has what kind of vegetation, and some similar data-maps, etc., equal-area is important.

But, other than that, because there’s no reason to expect area proportionate-ness in a flat map of a round Earth, then different magnifications isn’t wrong. That statement I’ve just made is a denial of a popular assumption, the assumption that area proportion relations different from those of the globe are wrong. They aren’t wrong unless the map’s area-relations are purported to match those of a globe.

As for shape-accuracy, that’s just a matter of the portrayal looking like what it’s supposed to represent. That’s a reasonable thing to expect and require in a map. But when we make rules about areas not being “too” far off, then we’re taking a whole different step in unjustified bossy-ness.

Look, say you have a wall on which you’ve posted lots of pictures. Pictures of people, animals, cars, and houses. Is someone going to say, “Oh, that’s grotesquely wrong—Your houses are shown at a very different magnification than your facial portraits!” ?

There’s a good practical reason for that. Just as there’s a good practical reason why many projections show some parts of the globe at higher magnification.

But what if your photo of a GTO has been 1-dimensionally expanded till it no longer looks like a GTO? Is it still a picture of a GTO? What will your neighbor say about it?

Yes, I know, a world map, unlike the photo-wall, could (but shouldn’t be) regarded as one accurate picture. Of course it isn’t, and shouldn’t be so regarded. Yes, when you took your photos, you yourself _chose_ the scale and magnification of each of your photos. But, didn’t you also make a choice of scales and magnifications when you chose your map projection? Yes, you chose your photos’ scales and magnifications to optimize them for the subject of the photo—while, a world map often or usually does its varied magnifying in a way that neither you nor your map-users have chosen as optimal. The map just magnifies as it magnifies, for reasons other than optimality for you or your map-users. But so what? As long as the capricious magnification isn’t done at the expense of the area or shape of the less magnified regions, then no harm is done. Some regions are serendipitously benefited as regards area and resolution. That’s better for some, and worse for none.

No harm done, unless you want equal-area, in which case you should instead use an equal-area map.

The point of my photo-wall comparison was merely to say that it’s nice if a representation looks like what it’s supposed to represent. Its magnifications are less important, to say the least. …unless you need the map to compare areas, in which case there are maps that are good for that purpose. NGS Winkel isn’t one of them.

I emphasize that this issue about areas is peripheral to the merit comparison of Eckert III and NGS Winkel, because Winkel only magnifies the Arctic by a factor of 2.58, compared to NGS Winkel’s 2.08

It’s merely a fault to put up with in order to gain other benefits.

One of the benefits that it gains is better magnification and resolution in the highly magnified regions. That isn’t a “fault” unless you expect or need area proportionate-ness. See above. In the case of Eckert III vs NGS Winkel, one of the benefits Eckert III gains is: It doesn’t mess up the Earth’s shapes in the very unusually widespread and extreme amount that NGS Winkel does.

You can pretend to make lemonade out of that lemon by exclaiming that it gives more room for examination, but unless the audience specifically wants that benefit for that region, then no, it’s just distortion and therefore unfortunate.

Call it what you want, but it isn’t a bad thing unless the map is needed for area comparisons. Yes you’re right—The audience didn’t ask for that magnification of particular regions, but does that make it bad? Remember that you agreed that Eckert III’s arctic magnification _doesn’t_ come at the expense of Ecuador’s size or shape. That better magnification and resolution of the arctic and Antarctic is un-asked-for, but it’s also free of charge. The Arctic serendipitously gets better magnification and resolution, because of geography, and constraints due to other desired map properties. What’s wrong with that?

We don’t make globes with some parts squished and other parts magnified, for rather obvious reasons.

Of course, because a globe is intended as a reasonably accurate model of the Earth, something that (the public know) is entirely impossible for a flat map, as you’ve pointed out.

I’m not saying that the arctic magnification of Eckert III and NGS Winkel is intentional. I’m saying that it’s a good thing for anyone wanting to closely examine the Arctic, and that it isn’t a bad thing for anyone—except for someone who needs an equal-area map for some specific purposes. I have nothing against atlases or wall maps being equal-area. Hammer-Aitoff and Quartic are among the maps that I advocate for those purposes. …as is Eckert IV, whose shape-distortions, unlike those of NGS Winkel, at least gain equal-area in a high-space-efficiency map, with continents looking relatively uniformly upright over a large range of longitude. I don’t like the shapes of NGS Winkel or Eckert IV, but at least Eckert IV does it for a good reason, and gets something (equal-area) for it.

…and, for that matter, Gall Orthographic gets something for it too: Equal area in a map with space-efficiency of unity, and which shows _all_ longitudes uniformly upright.

But, though Eckert IV’s shape-distortion is tolerable (because it’s for a good reason), Gall-Peters’ shape distortion is just too much, which is why I myself wouldn’t wall-post it. Anyway, a main value of equal-area is that it’s equally easy to show and find things everywhere. But there’s some question about that being so, when there’s such an extreme directional disparity of scale.
I suppose that writing could be aligned north-south in the tropics, making good use of the N-S expansion. …and that the N-S displacement, by increasing the N-S map-distance between nearby points, also keeps them from getting in eachother’s way and competing for E-W space too. So it could help to ease E-W space-clutter, as well. So maybe Gall Orthographic _does_ give the topics genuine clarity-equality via its equal-area. But, any resulting ease of finding things must be reduced some by the dramatic change in directions and the scale-disparities, and that’s a big disadvantage of Gall Orthographic, in addition to its awful looks.

So in NGS Winkel there are some European countries with good shapes and directions. …which, in NGS Winkel, certainly can’t be said for the U.S., or continents other than Europe. …or ¾ of the Earth’s surface.

Despite your insensitivity to distortion on some projections, all projections have this problem.

Oh, not at all, not to that humungous degree, even at the central meridian. Not every map shows ¾ of the Earth so drastically shape-distorted. Very few do. Only NGS Winkel, Peters, and a few of Peters’ close relatives. (and maybe a few other rarely-encountered projections).

I don’t think disagreeing preferences can be called prejudice

That depends upon whether the person recognizes that “his” preference represents an æsthetic choice from amongst many competing, inaccurate representations, or whether instead the person believes his preferred representation is the accurate one. In the latter situation, the preference is a prejudice.

Anyone would be very mistaken to believe that a flat world map is genuinely fully accurate.

In your case, I would suggest some prejudice mixed with a lot of preference, with prejudice arising out of insensitivity to distortions that trained others recognize readily when they see them. That insensitivity extends well beyond the natural perceptual limits of about 5% error.

You mean because I don’t recognize the badness of magnifying the Arctic by 2.58 instead of just 2.0? :^) Would your trained experts notice the _widespread, blatant and unusual_ NGS Winkel shape-distortion that I keep mentioning? As for the tradeoff and balance between those, see above.

But, instead of answering the charge of prejudice, I’d rather just stick to the subject and its objective facts…And isn’t that exactly what prejudice keeps us from?

I don’t agree that people need to be educated to different preferences.

It’s not a matter of educating people to different “preferences”. It’s a matter weaning them off of any particular false belief about how things look and how projections work. If people’s exposure to world maps is never dominated by any particular projection or presentation…

…such as the ubiquitously predominant Robinson and NGS Winkel?

, false beliefs about how things look can’t ever gel. Instead they’ll (at least subliminally) recognize that they all look different because none of them are right and none of them can be right. Possibly some “average” view of all them will impose itself into their consciousness, and that average view should be closer to a globe’s view than any single narrow class of projections would afford them.

Fair enough, but anyone _who is interested in or needs_ area comparisons, is surely interested enough, and either is or will soon be well-informed enough, to know about the area distortions on the map s/he is using, which are important to hir purposes. Anyway, s/he will surely choose an equal-area map that area-important hir purpose.

And I remind you that, regarding arctic magnification, for the purpose of Eckert III vs NGS Winkel, we’re comparing arctic magnifications of 2.58 and 2.08

For example, if maps other than Robinson and NGS Winkel are convincing people that the Arctic is a lot larger than it is, with serious adverse environmental consequences, and if that harm can be attributed to the maps, then you could say that preference for those maps is undesirable in terms of its results, and is a result of not considering the necessary information. But has it really been shown that that’s so?

No, it hasn’t, and that’s not my point. My point is that you cannot make credible claims by discounting concerns you’re not troubled by.

But I remind you that I clarified that this isn’t social protest. I _do_ discount concerns that I’m not troubled by, and I’ve been telling you why, and re-assuring you to likewise not be troubled by them. It’s fair for me to do that, because I admit that it’s just my own evaluation --while also recommending it as your evaluation too, and telling you some merits of it.

But I’m not trying to change your opinion. I’m merely making observations about a subject that I like. No animosity or anger, or goal-orientated-ness.

My position is that you can use any “reasonable” projection more or less responsibly, where responsibly means you recognize the full gravity of what you have sacrificed in order to attain what you favored.

Of course. And, additionally, maps aren’t just for “use”. I regard world map projections as an art-form. Aesthetics is a consideration for evaluating world maps, and that’s another good reason why things should look like what they’re supposed to represent (unlike ¾ of the Earth in Winkel).

You say that I ignore where other maps fail just as badly in that regard.

Eckert III? I admit that Eckert III doesn’t portray Antarctica (1/36 of the Earth) as well as Winkel. But 1/36 isn’t the same as ¾, is it.

Hammer-Aitoff? As I said, its distortion near the outer meridian realistically resembles the perspective foreshortening on a globe, and thereby just enhances the maps globular realism.
No aesthetic problem there.

Arctic magnification? You mean 2.58 as opposed to 2.08?

No, Winkel objectively loses, in comparison the the other abovementioned projections, and that’s so without ignoring putative equally bad failures of other world map projections. The fact is that nothing except Peters and a few of its relatives does as bad as NGS Winkel.

As for the Arctic, it’s mostly ocean, and people aren’t so conscious of shape-errors there. Greenland? Sure, Winkel shows its shape a little better, but I wouldn’t go so far as to say that Eckert III is grossly unaesthetic there. Do you seriously claim that Greenland’s shape on Eckert III looks worse or stands out more than South America’s, the U.S.’s, and Africa’s shape on NGS Winkel?

Antarctica? Yes, as I’ve admitted above, NGS Winkels shows Antarctica much better than Eckert III does. Antarctica is about 1/36 of the Earth’s surface. Compare that to ¾ of the Earth’s surface, drastically FUBAR distorted by NGS Winkel.

The fact is, you give up some things that are important to some concerns no matter what projection you use

But, as described above, with Eckert III, you don’t uglify and mis-represent how the continents look as much as Winkel, except when displaying Peters or its close relatives.

No, it _isn’t_ all relative, or an even trade-off. NGS Winkel is a mess, for the reasons stated above, compared to Eckert III, Times Winkel, Oxford Winkel, Hammer, and Quartic.

, and just as importantly, whatever distortions are present in the projection will mislead all but the most highly trained audience. You cannot predict the consequences of that.

People will think that the Arctic is 2.58 times as big as it really is…instead of just 2.08 times as big as it really is?

with respect to the arctic,

Eckert III gives the impression that there is a lot more arctic than there actually is
And with respect to Greenland,

…To me the relative inflation of Greenland on the Eckert III is huge, and even the inflation on the Bartholomew (Times) version of Winkel tripel is blatant compared to Winkel’s preferred version of his tripel projection.

And yet you keep harping on about the arctic as a whole when I specifically wrote Greenland

I thought you said “It isn’t just Greenland. It’s the whole Arctic.” Then you told of various reasons why the Arctic (and Antarctic) as a whole is important. And so, instead of determining the projections’ magnifications of Greenland, I determined their magnifications of the Arctic.

But if Greenland is more important to you now (contrary to what you said earlier), then I say that Greenland’s N-S extent isn’t too different from that of the Arctic. Sure, Greenland doesn’t extend to the North Pole, and it extends a little south of the Arctic Circle. So we can expect Greenland magnification to be somewhat less than Arctic magnification. But now are you saying that 1) Greenland is more important than the overall Arctic and Antarctic; and 2) The comparison of Eckert III’s arctic magnification to that of NGS Winkel is worse than the comparison of their Arctic and Antarctic magnifications? Because we’ve now found out just how much Eckert III and NGS Winkel magnify the arctic, then now it’s suddenly about Greenland instead of the Arctic and Antarctic?

But yes, Eckert III inflates the arctic a quarter more than Winkel tripel, and yes, that’s significant and obvious.

Incorrect. By your own reported arctic magnifications, Eckert III magnifies the Arctic about 1.2435856 times as much as NGS Winkel does. Or call it about 1.24 …not “more than a quarter more”.

If one projection didn’t magnify the Arctic at all, and another magnified it by a factor of 1.24, then that might be significant. But, when one projection already magnifies the Arctic by a factor of 2.08, and the other magnifies it by a factor of 2.58, then that isn’t the same thing at all—even though the 2nd projection still magnifies 1.24 times as much as the 1st one.

Why aren’t [NGS Winkel and Robinson stocked in stores? Because no one wants them. …Ok, I was just trying to guess why nearly _all_ atlases use Robinson as their main world map. It seemed to me that the best explanation was that cartographers advise them to. Maybe it’s coincidence. Or maybe publishers got together and took a vote on what world map projection they should all adopt as a group.

…the answer is far more likely to reside in the fact that publishers rushed to follow National Geographic into the Robinson projection because of the fanfare National Geographic made out of their switch in 1988. Whereas, when National Geographic switched to Winkel tripel, they said nary a word about it. Meanwhile publishers had invested in Robinson and probably don’t see any benefit to retooling their cartography to the Winkel tripel. Some of them probably aren’t even aware that National Geographic switched. National Geographic sells millions of maps. If they were losing sales because of their projection, they’d be switching.

Yes, that sounds like a much better explanation for Robinson’s continuing wide use in atlases.

But it doesn’t answer my other question and statement: “Why aren’t NGS Winkel and Robinson stocked in stores? Because no one wants them.”

The simple fact is that the buying public is nowhere near sensitive enough to differences between Robinson and Winkel tripel to affect sales.

Yes, I feel that the reason why no one objects to NGS Winkel, in National Georgraphic magazine and other NGS publications, is simply because no one notices or cares enough to make it a consideration in the purchase of an issue of National Geographic, or some other NGS publication. That projection pleases NGS, and no one else notices or cares.

…given that those preferences are mere prejudices, their relevance to a responsible choice of map projection is questionable.

Let’s not fall into the prejudice of assuming that preferences different than our own are prejudices.

Projections’ merits can be compared by comparing their advantages and disadvantages, in comparison to eachother. …instead of questioning others’ judgment or seeking to discount other preferences by calling them “prejudices”.

[P]andering to people’s prejudices for any specific projection is a questionable way to choose a projection, whereas a responsible way is to disabuse people of their prejudices. That is, to educate them.

You’re assuming a lot there.

And what does NGS Winkel educate people to, in comparison to Eckert III? …that the Arctic is only 2.08 times as big as it really is, instead of 2.58 times as big as it really is? …that a full ¾ of the entire Earth is drastically east-west compressed, in comparison to its actual shape?

Yes, NGS Winkel _does_ educate people to a more realistic shape for Antarctica. But, as I said, Antarctica is only 1/36 of the Earth’s surface, compared to the ¾ of the Earth that NGS Winkel makes a FUBAR mess of. What kind of education is that?

Over most of the Earth, Eckert III’s shapes of continents and countries is more accurate than that of NGS Winkel. How’s that for education? Might people not wonder why NGS Winkel’s continents look so different than those on the globe? …unnecessarily different, because, over most of the Earth, they aren’t like that, to that extreme degree, on Eckert III.

Sure all flat world maps distort, but when that distortion is specified and known, how bad is it really?

Really bad. The general public has little sense for the distortions.

So teachers can mention it to their students, when they display maps and discuss geography. Parents can likewise mention it to their kids. Atlases could indicate their maps’ distortions (in fact they often do, in a preliminary section on map projections).


.

I’m not arguing against world maps. I’m arguing against rubbing the public’s nose into any restrictive set of projections.

Yes, it would better if people didn’t just encounter, for the most part, the same one world map projection in atlases. …and another one very nearly exclusively displayed in the National Geographic Society’s publications.

I’m also arguing against personal biases being used as justifications for projection X and against projection Y when there are objective benefits to both and severe problems with both.

I’ve described the severe problems of NGS Winkel, and, in this message, I specifically ask you to specify what benefits NGS Winkel offers to balance and justify its severe distortion that I’ve described. …And what severe problems of Eckert III, Times Winkel and Oxford Winkel are you referring to that are as bad as those of NGS Winkel?

I’m asking for specifics.

I’ve told how the shape distortions near the outer meridians of Hammer and Quartic realistically resemble the perspective foreshortening at the edge of a globe, thereby only enhancing those projections’ realistic globular appearance. ..and therefore aren’t an aesthetic problem. So therefore I add Hammer and Quartic to the projections that are objectively not as bad as NGS Winkel. (…and of course they don’t have NGS Winkel’s 2.08 magnification of the Arctic)

Hammer-Aitoff:





Quartic Authalic:




Michael Ossipoff

[daan]

1. Regarding Robinson, it would be nice to disclose a map’s projection. It would be nice for a “map” to be a specified mapping of lat,lon to X,Y.

Straw man. This was never in dispute. Your thesis was that Robinson could not even be called a map.

2. Regarding NGS Winkel, it would be nice for representations to look like what they’re supposed to represent.

Regarding any projection, it would be nice… but impossible.

Yes, that’s what I meant. For anyone studying the Arctic, in regards to the things you mentioned, when area-proportionality matters, NGS Winkel wouldn’t do. Equal-area would be needed. But otherwise, for general purposes, 2.58 arctic magnification isn’t importantly more than 2.08.

This comparison is wrong, and it’s one more reason in a long line that I have pointed out along the way that makes this debate ceaseless and unproductive. You compare the wrong things and you ignore what does not suit your rationalizations. Those ratios are with respect to the nominal scale. That’s not what’s important. What’s important is how those ratios compare against the entire rest of the world. Your touted Eckert III inflates the arctic by 2.58 over an equatorial region that’s only 73%, on average, of the size that it should be between the tropics, for a total ratio of 3.53. Meanwhile Winkel tripel’s tropical deflation is at 83%, for a total ratio of 2.51.
For this disinterested (but nonexistent) reader, here is what’s going on. The first image is an equal-area map. That is, everything has the right proportions, but in order to achieve that, shapes suffer. The projection is Eckert IV.

Eckert IV

Eckert IV map.jpg (103.09 KiB) Viewed 2452 times

Notice how much of the map is taken up by Africa and South America. Referring to a globe or maps of those continents specifically, notice how ridiculously stretched Africa and South America are.

RogerOwens prefers this representation, which is Eckert III.

Eckert III

Eckert III map.jpg (105.57 KiB) Viewed 2452 times

Sure enough, as advertised, Africa’s shape looks about right. Refer to a globe or to a map of (just) Africa to confirm. However, notice several things. Notice that Asia looks way stretched east-to-west, as does Canada and Alaska, not to mention Antarctica. But most of all, look how ridiculously puny Africa and South America are compared to the map as a whole.

RogerOwens has been arguing prolifically against this map, which is Winkel tripel as formulated by Winkel. It’s what National Geographic Magazine has been using regularly for world maps since 1998.

Winkel tripel

Winkel tripel map.jpg (113.66 KiB) Viewed 2452 times

Notice that Africa and South America take up more of the map than the Eckert III map’s versions do, but less of the map than Eckert IV does. Notice that, in order to pay for this improvement in proportion, Winkel stretches Africa and South America out, but not as much as Eckert IV. Notice that Canada and Asia look closer to their actual shapes than either of the other two maps, particularly because of the less severe stretching east-west.

Now, each of these maps is ridiculous. None of them looks like reality. However, each of them has its place, and where one map excels, another fails, and there is no free lunch to be had. RogerOwens wishes you to subscribe to his belief that Eckert III is less ridiculous than Winkel tripel. I say piffle. I say RogerOwens’s fixation on certain kinds of distortions while discounting others amounts neither to a keen sense of geography nor a keen sense of æsthetics. At best it’s merely personal preference, and all these thousands of words expended in justifying that preference carry no more persuasion for me than someone going on ad nauseam about why an SUV is better than a family sedan.

To see exactly where these trade-offs happen and what their impact is, take a look at these distortion diagrams:

Winkel tripel Inflation/deflation

Winkel tripel areal.jpg (92.04 KiB) Viewed 2452 times

Eckert III inflation/deflation

Eckert III areal.jpg (93.07 KiB) Viewed 2452 times

Here you can see where the two projections increase or decrease sizes. White is the region where sizes are about right compared to the stated scale of the map. In this representation, a 20% shrinkage (for example) has the same tone as a 20% inflation. Toward the center, both projections shrink things, but Eckert III shrinks them a fair bit more than Winkel tripel. Toward the poles, both projections inflate things, but Eckert III inflates them a fair bit more. Winkel tripel represents a much greater swath of the globe with something close to proper proportions.

The next diagrams show “warping”. Again, the white areas are regions of little deformation. The more magenta, the greater the deformation.

Winkel tripel deformation

Winkel tripel angular.jpg (104.42 KiB) Viewed 2452 times

Eckert III angular

Eckert III angular.jpg (96.55 KiB) Viewed 2452 times

Eckert III favors the tropics at the expense of the rest of the map (hence Africa and South America look pretty good). Winkel tripel, on the other hand, distributes moderate distortion across a wider swath of the surface, resulting in most places looking somewhat wrong. Overall Eckert III has less angular distortion across the subarctic landmasses, just as Winkel tripel has less area disproportion across the subarctic landmasses.

I’m not saying that the arctic magnification of Eckert III and NGS Winkel is intentional. I’m saying that it’s a good thing for anyone wanting to closely examine the Arctic, and that it isn’t a bad thing for anyone—except for someone who needs an equal-area map for some specific purposes.

And I could just as well argue that Africa being stretched isn’t a bad thing for anyone—except for someone who needs shapes to be correct for some specific purpose. You’ve simply decided that shapes up to the mid latitudes are important and nothing else is except when there’s a specific need, and you obdurately go on and on and on and on about it. I am not going to debate this any further. It’s pointless. What you express is a bias. To the disinterested (but nonexistent) reader, I encourage you to be cognizant of the tradeoffs in map projections. Treat anything you give up with respect.

Would your trained experts notice the _widespread, blatant and unusual_ NGS Winkel shape-distortion that I keep mentioning?

Of course. It‘s amusing you’d speculate otherwise. And unlike you, they also notice the widespread, blatant and unusual Eckert III misproportions I point out with ever greater specificity, whose magnitude and consequences leave you blissfully untroubled.

…such as the ubiquitously predominant Robinson and NGS Winkel?

Why do you continue to harp on this? It’s not a point of contention.

I _do_ discount concerns that I’m not troubled by, and I’ve been telling you why, and re-assuring you to likewise not be troubled by them. It’s fair for me to do that, because I admit that it’s just my own evaluation --while also recommending it as your evaluation too, and telling you some merits of it.

And I’m reassuring you that you should be troubled by what you’re discounting. They are important concerns. You’re not going to argue me into sharing your preferences because I think those preferences are naïve.

Aesthetics is a consideration for evaluating world maps, and that’s another good reason why things should look like what they’re supposed to represent.

Apparently you cannot grasp that some people think Eckert III’s representation of proportions looks ridiculous. The projection does not make things look like what they’re supposed to represent. In isolation it makes South America and Africa look decent. The rest of the world is pretty much a mess, and they’re even more of a mess because the very shapes that look good don’t fill up nearly as much of the map as they ought to, leaving the global proportions misshapen and ridiculous. I get that you don’t care, but not caring doesn’t justify the enormous amount of effort spent attempting to justify not caring.

Eckert III? I admit that Eckert III doesn’t portray Antarctica (1/36 of the Earth) as well as Winkel. But 1/36 isn’t the same as ¾, is it.

Well great; let’s just use Mercator! No matter that it portrays Antarctica as infinite in extent; Antarctica is only 1/36th!

But, as described above, with Eckert III, you don’t uglify and mis-represent how the continents look as much as Winkel,

This is just wrong. Eckert III mangles practically all of Canada, northern Europe, and northern Asia worse than Winkel tripel. You go on and on about east-west compression on Winkel tripel, but those regions are stretched east-west on Eckert III. It’s fairly mysterious how you do not see that, but regardless, it’s right there in the distortion diagrams, and certainly I can see it quite obviously—just as I can see the markedly worse proportions in Eckert III. While average angular distortion for the two maps is similar across those regions, Winkel tripel wins strongly in those same regions by preserving areas across large swaths. In contrast Eckert III deforms large regions because of the change in proportionality. To me Eckert III uglifies just as much as Winkel tripel, so it’s pointless to make those ostensibly æsthetic observations as if they are obvious and given. They aren’t. They are near the heart of the dispute.

I thought you said “It isn’t just Greenland. It’s the whole Arctic.” Then you told of various reasons why the Arctic (and Antarctic) as a whole is important. And so, instead of determining the projections’ magnifications of Greenland, I determined their magnifications of the Arctic.

But if Greenland is more important to you now (contrary to what you said earlier),

Just hopeless. You’re mixing up two different paragraphs. I wrote about the arctic as a whole:

It isn’t just Greenland, and it isn’t just “slight”. It’s the entire arctic and antarctic. Compared to Winkel tripel (Winkel parameterization) or Robinson, Eckert III gives the impression that there is a lot more arctic than there actually is. Maybe you only care about inhabited land masses. Maybe a lot of people, such as those learning about global warming or about whale habitats or about cod fisheries or about solar radiance reflectivity, actually care about the proportions and impact of the arctic and antarctic, including the oceans. Do I think there’s anything particularly wrong with Eckert III? No! Do I think Eckert III is better than anything else across a broad range of concerns? No! Like any projection, it’s horrible as a world map, but like other reasonable projections, it’ll do once in awhile for some purposes.

and I wrote about Greenland specifically:

I cannot figure out what you’re saying here, but to be clear, to me the relative inflation of Greenland on the Eckert III is huge, and even the inflation on the Bartholomew (Times) version of Winkel tripel is blatant compared to Winkel’s preferred version of his tripel projection. If those differences mean nothing to you, fine, but your repeated claims that your preferences are norms of some kind just don’t mean anything to me.

They are in two different paragraphs. Yet when you responded, you responded to my Greenland comment as if it were my arctic comment:

but to be clear, to me the relative inflation of Greenland on the Eckert III is huge...

Eckert III’s arctic magnification factor is 2.58 as opposed to NGS Winkel’s 1.9 Would you call that a “huge” difference in arctic magnification factors?

Why are you asking about the arctic when you yourself excerpted from my Greenland comment?

, and even the inflation on the Bartholomew (Times) version of Winkel tripel is blatant compared to Winkel’s preferred version of his tripel projection.

More than Eckert III’s 2.58 magnification of the Arctic?

Why are you asking about the arctic when you yourself excerpted from my Greenland comment?

If those differences mean nothing to you, fine

The difference between 2.58 and 1.9 doesn’t mean much to me. Does it to you?

Why are you asking about the arctic when you yourself excerpted from my Greenland comment?

But yes, Eckert III inflates the arctic a quarter more than Winkel tripel, and yes, that’s significant and obvious.

Incorrect. By your own reported arctic magnifications, Eckert III magnifies the Arctic about 1.2435856 times as much as NGS Winkel does. Or call it about 1.24 …not “more than a quarter more”.

I did not write more than a quarter more! With this sort of misquoting going on and on and on, the conversation cannot progress.

Yes, I feel that the reason why no one objects to NGS Winkel, in National Georgraphic magazine and other NGS publications, is simply because no one notices or cares

…and the ones who do notice and care disagree with you that it’s a poor choice.

Over most of the Earth, Eckert III’s shapes of continents and countries is more accurate than that of NGS Winkel. How’s that for education?

It’s horrible for education because it’s factually wrong and because it ignores the other half of the equation: proportions across the map.

Anyway, seriously, I’m done with this topic. Peace.

— daan

[RogerOwens]

I don’t agree that this discussion hasn’t gotten anywhere. We’ve arrived at much agreement and consensus regarding the relative merits and advantages of Eckert III and NGS Winkel.

Subtopics in this post:

(not necessarily in this order)

Poll results
Replies to some of daan’s comments
Actual and portrayed areal relations in Eckert III & NGS Winkel
Shapes-comparison over percentages of the globe
Worst-shape comparison (It’s a tie)
Worst-shaped-continent comparison
Balance in compromise-maps
Desirability (and reason for lack) of publishing a variety of projections
Area vs shape, practical & aesthetic, objective facts to guide aesthetic choice
The role of fashion
Additional projections better than Winkel and Robinson

When, in this post, I say “Winkel”, I refer to NGS Winkel Tripel. When, in this post, I say “Eckert”, I refer to Eckert III.

I started some polls, showing images of Eckert and Winkel, and asking which projection looks best, and more accurately and globe-like portrays the continents. The number of people preferring Eckert to Winkel exceeds the number preferring Winkel to Eckert by a factor of about 1.5

The poll announcement and description didn’t include any advocacy, arguments or description by me—just the images. A general audience was polled—people who couldn’t be expected to have any particular prior opinion.

Shape-comparison and Area-comparison:

I’ll separately answer daan’s area comments and shape comments. Shape will be first, because shapes are the advantage that I claim for Eckert, vs Winkel.

Shape:

Immediately below is probably the best statement of daan’s to answer, in that regard:

In isolation [Eckert III] makes South America and Africa look decent. The rest of the world is pretty much a mess…

For one thing, that obviously isn’t true. England and Spain, about 90 degrees from the central meridian, look fine in Eckert. …a lot better than Winkel shows the U.S., at the same longitude-distance from the central meridian. The outermost mid-latitude graticule-quadrangles in Eckert look close to globe-like (in terms of fat vs skinny), and noticeably better than they do in Winkel.

Let’s look at latitude-bands;

When I name a latitude or a latitude-band, I’m referring to it and its negative too.

The 0-30 latitude band:

That’s the latitude-band across which Africa pretty much extends. It also encloses most of South America. So, given daan’s comment above, it’s a good place to start.

Yes, in latitude-band 0-30, Eckert definitely looks better than Winkel—no contest.

And the 0-30 latitude-band encompasses half of the Earth.

Therefore in order for Winkel to have better shape-portrayal over as much of the Earth as Eckert, then Winkel would have to be better over the _entire_ other half of the Earth. The entirety of the Earth pole-ward of lat 30. Is it? Of course not:

Looking at 30-degree graticule quadrangles in the lat 30-60 range:

Yes, Winkel’s graticule-shapes are globe-like near the central meridian, and Eckert’s are a bit fat there. But near the outer meridian, Eckert’s graticule shapes are more globe-like than those of Winkel, which get far too skinny there.

So Winkel doesn’t win the half that it needs, in order to not lose to Eckert in most of the Earth.

So Winkel obviously doesn’t have better shapes in as much as half of the map, compared to Eckert. Winkel loses to Eckert in the comparison of which map is better than the other over a greater percentage of the Earth’s surface.

The U.S. latitude-band (lat 30-49):

That latitude band comprises about ¼ of the Earth (as I said, when naming latitudes or latitude-bands, I refer to both their positive and negative values)
.
In a Winkel map that shows the eastern side of the U.S starting around 80 degrees west of the central meridian, the U.S. is drastically, seriously, east-west compressed. Shall we generously say that Winkel is ok out to 90 degrees from the central meridian? In an Eckert map that shows Britain and Spain about 90 degrees east from the central meridian, Britain’s and Spain’s shape looks fine, in regards to fat/skinny.

And, as mentioned above, towards the outer meridian, Eckert’s mid-latitude shapes are better than Winkel’s.

From the above paragraphs, we can say that, roughly, Winkel gets bad by 90 degrees out (from the central meridian), but Eckert, though a little fat near the middle, begins to look fine by 90 degrees out., and then remains better than Winkel.

The lat 30-49 band comprises about ¼ of the Earth. So, if we, roughly, say that Winkel has better shape in half of that band, and Eckert has better shape in the other half of that band, then, in terms of fractions of the earth, that band gives 1/8 to Winkel, and 1/8 to Eckert.

Eckert wins in the lat 0-30 half of the Earth. Let’s (maybe generously) say that Eckert wins everywhere in the lat 49-90 quarter of the Earth.

So, as a rough estimate, based on the above estimates, Eckert beats Winkel in 5/8 of the Earth.

The part of the Earth in which Eckert has better shapes than Winkel is 5/3 as large as the part of the Earth in which Winkel has better shapes than Eckert. …almost twice as large.

But, even if you contest that 5/3 figure, which depends on saying that it’s an exact draw in the lat 30-49 band, it’s still undeniable that Eckert wins in the lat 0-30 half of the Earth, and that Winkel loses in part of the other half. …meaning that Eckert’s winning Earth % is greater than that of Winkel.

And Winkel’s (smaller) better-region includes the Arctic and Antarctic, two areas that are considerably less-often referred to when people refer to a map.

So, in terms of the % of the Earth in which one of the two projections has better shape, Winkel undeniably loses to Eckert.

Of course someone could say that they’re more concerned or influenced by the bad-ness of a projection’s _worst_ shape-distortion, regardless of the two maps’ better-ness % of the Earth.

So then, which projection is worse, at its very worst? It’s a tie. Both Eckert and Winkel have infinite east-west scale at the poles, with the shape-distortion that goes with that. And, over finite-size polar regions, they both have east-west scale that is increasing without bound as the pole is approached

More about that later, when areal infidelity is discussed.

Ok, but which projection has the worst shape for a continent? It must be admitted that Eckert does. Eckert’s Antarctica is considerably worse-shaped than Winkel’s Africa is.

Eckert’s and Apianus II’s Antarctica looks like some malevolent octopus lurking at the bottom of the world. So what? It isn’t a globe, and it can’t all look like one. (…not if you want a flat pole, anyway).

Winkel replaces a really bad shape with somewhat bad shape over most of the Earth.

Speaking for myself, I’d rather have the shape-distortion concentrated at one small, less often referred-to place, instead of smearing it all over the map, as Winkel does. Evidently most people agree with me.

Besides, asking that a polar continent be shown with good shape is asking too much. Achievable, but still unfair to demand, when other, more important and desirable, properties must be traded for it.

Because of Eckert’s bad-shaped Antarctica, Winkel would admittedly win that worst-continent comparison (…if we pretend to be unaware of both maps’ infinity-approaching shape distortion as we near the poles—a big “if”).

One more thing: Winkel’s extreme low-latitude distortion could be excusable, could be justified, if it bought good shape in mid-latitudes. But it doesn’t, does it. Look at Winkel’s U.S. Mid-latitude shape is good only near the central meridian.

That’s why I say that Winkel is a lose-lose proposition.

Away from the central meridian, everything quickly goes south.

Areas:

When I say “Band-Area-Ratio” (BAR), I refer to:

(arctic area+antarctic area)/(tropical area).

To refer to the factor by which a map’s BAR differs from that of the globe (the map’s BAR divided by the globe’s BAR), I’ll say “Band-Area-Ratio Falsification-Factor” (BARFF).

Reasonably and fairly, daan chooses BARFF as his preferred way to judge arctic magnification. He points out that, while Eckert’s BARFF is 3.53, Winkel’s BARFF is “only” 2.51. :^)

Winkel’s BARFF is 11% greater than that of Apianus II, and incomparably greater than that of Aitoff. And of course Hammer and Quartic have no such falsification (Their BARFF = 1), because they’re equal-area projections.

Let the reader decide for hirself if the above-stated difference in the BARFF of Eckert and Winkel justifies Winkel’s surrealist, Dali-esque shape-distortion, shown in the images that daan posted.

But there’s another kind of areal infidelity that daan seems to have quietly tiptoed around: In pseudocylindrical projections like Eckert or Apianus II, every graticule-quadrangle between the same pair of parallels has the same area, just as it does on the globe. Not so with Winkel. On Winkel, even between the same pair of parallels, the graticule-quadrangles at the outer meridian are sometimes noticeably larger than those at the central meridian.

So I’d like to suggest one possible measure of that kind of areal infidelity: The greatest factor by which the area of any 10-degree graticule-quadrangle at the outer meridian exceeds that of the 10-degree graticule-quadrangle at the central meridian in the same 10-degree latitude-band.

I’ll call that “Longitudinal Areal Falsification-Factor” (LAFF).

Winkel has noticeable LAFF. But LAFF does not occur in Eckert III, Apianus II, or in any pseudocylindrical projection.

I don’t object to LAFF, or areal-infidelity in general, in Lagrange and August, where conformality is chosen instead of areal-fidelity. I’m just saying that acceptance of Winkel’s LAFF, when Eckert III and Apianus II don’t have any LAFF, is inconsistent with using areal-fidelity as an advantage of Winkel over Eckert.

Speaking of Apianus II, let me define it. It’s simply the pseudocylindrical projection (straight horizontal parallels, each with its (usually different) constant uniform scale) with parallels equally-spaced, and with equator and central meridian having the same scale. Constructed in an ellipse.

Winkel has 11% more BARFF than Apianus II does.

Winkel has noticeable LAFF. LAFF doesn’t occur in Apianus II or Eckert III.

Could it be that “complicated” isn’t always “better”?

And, just as with Eckert, Apianus II doesn’t share Winkel’s widespread funny E-W compression. Low latitudes (lat 0-30), where Winkel is at its worst, worse than Eckert III or Apianus II, comprise half of the Earth’s surface.

Another difference: Winkel (like all flat-polar projections, including Eckert III and Robinson) has infinite east-west scale at the poles. Approaching the poles, the east-west scale increases without bound, and so is also very large in a finite-size polar region. Defenders and apologists of Robinson and Winkel are all-concerned about areal infidelity, but don’t seem to mind the east-west scale increasing without bound near the poles. It’s rather like someone boasting about their meticulous housekeeping, while pretending to not notice a large dog-dropping in the middle of the living-room carpet.

Arbitrarily large scale-exaggeration is ok, as long as it’s only in one dimension? Is that a bit hypocritical? An inconsistent choice in the service of a pre-decided fashion preference?

I’m not saying that the arbitrarily-increasing scale approaching a flat pole bothers me. I’m saying that acceptance of it is inconsistent with concern about area-magnification. Better areal fidelity…but scale-error without limit? Talk about cherry-picking.

daan, reasonably and fairly, evaluates and compares conformal maps by their maximum scale divided by minimum scale. But does he look the other way, for the fashionable Robinson and NGS Winkel compromise-maps, whose max/min scale ratio is infinite? Compromise can be a funny thing. Fashion can be a funny thing.

If it weren’t for Winkel’s better portrayal of Antarctica’s shape, maybe we could say that Apianus II almost completely dominates Winkel. Antarctica is 1/36 of the Earth.

If someone’s goal is to minimize the distortion of the worst-shaped continent, then of course Winkel beats Eckert and Apianus II. (which is why Winkel won by G&G’s formulas) But (as mentioned above) I’m in the habit of forgiving bad-shaped Antarcticas, because of Antarctica’s small size, and less-frequent reference-need, and because of the desirable properties that can be traded-for. Not everyone agrees with me on that. Some prefer Winkel to Eckert because they don’t forgive Eckert’s Antarctica. Of course no one’s wrong about that, and I respect their preference.

…but evidently the number of people who agree with me on that is about 1.5 times as great as the number who don’t, based on recent poll results.

Another reason for preferring Winkel is its realistically converging meridians. But, whether we want good Antarctica shape, or converging meridians, those things can be gotten, more-so, with Aitoff, Hammer, or Quartic—without Winkel’s funny N-S/E-W scale-disparity. And of course, with Hammer or Quartic, you get equal-area.

For looks, if you demand good Antarctica shape too, maybe Aitoff is unbeatable. But Hammer’s appearance is nearly as good (you have to look for the difference), and it’s equal-area.

Because Aitoff is so good, the averaging of Aitoff with square-grid Cylindrical-Equidistant sounds like a fine idea. It looks good too. The Oxford Atlas uses it. It’s Oxford Winkel. Due to its Aitoff parentage, its Antarctica looks pretty good—much better than that of Eckert III. But Oxford Winkel loses Eckert’s simplicity, linearity and complete absence of LAFF.

But, again, if Oxford Winkel’s abovementioned advantages are important, then why not just use Aitoff, Hammer, or Quartic?

I don’t know the initial motivation for Quartic, but maybe someone wanted to make Hammer pseudocylindrical. It’s written that they started with Hammer’s parallel-spacings, and extended each parallel outward enough to make the map equal area. …and then horizontally expanded the map, just a little, to optimize shape.

The result is an equal-area map that resembles Hammer, but has the convenience of pseudocylindrical-ness. And it results in a quartic curve for the map’s boundary, hence the name.

Quartic doesn’t look quite as globularly-realistic as Hammer, but it has a nice stylized look—both in its overall boundary-shape, pointed poles, and Antarctica-shape.

Unavoidably, the pseudocylindrical-ness increases shear-distortion some.

Apianus II, like Eckert III, isn’t complicated enough to be liked by cartographers. And Apianus II was introduced in 1524—no novelty-points there. Elliptical world maps have been used a lot, and so they can’t qualify as the latest fashion. Of course the law of fashion is change.

The current vogue is compromise. …evidently even to the extent that flat-polar, part-oval, hybrid-boundary maps are the only permissible world maps. daan is right to say that variety is desirable. But atlas-publishers probably are sensitive to cartographers’ criticism of maps that aren’t compromising enough. Hence the complete lack of variety.

Here’s what Eckert III is:

Like Apianus II, Eckert III is a pseudocylindrical with equally-spaced parallels and with equal scale on the equator and the central meridian. But, instead of an ellipse, Eckert III is constructed in a figure with convex semicircular right and left sides. At top and bottom, Eckert III has horizontal flat-pole-lines extending over the inner half of the map’s width--one pole-line connecting the tops of the semicircles, and the other connecting the bottoms of the semicircles.

Of course fashion continually changes in lots of areas. I’m not singling out cartographers in that regard. “Out with the old, and in with the new!” What people already like, or what has previously been in use, is by definition, out of date, and is claimed to represent a need for re-educating the public. Certain professions seem to graciously accept the mandate and the mantle to educate the public to their current fashion regarding map projections.

Aside from the matter of % of the Earth portrayed better-shaped in Eckert, vs Winkel, I want to repeat this consideration: In the low-latitude zone, half of the Earth, where Eckert does considerably better, and where Winkel does considerably worse, lie Africa and South America.

Those are the front-and-center, right-in-front-of-you, continents. They’re your first impression on a world map. They’re the continents that are relatively free-standing and small enough so that there’s a definite obvious particular shape that they should have on a flat map (in contrast to Eurasia or the Pacific Ocean, which are too large to have an expected meaningful shape on a flat map). You can map South America and Africa on flat maps of sufficiently small region that the shape means something, and you expect it to be a certain way. You can easily observe Africa’s and South America’s shape on a globe.

That’s why those two relatively-small, relatively free-standing, prominently-placed continents give people their first impression of a map’s accuracy and appearance. It’s why their distortion is considered particularly objectionable.

daan speaks of “balance”. But balance should take into account the matter of which errors are more noticeable, more objectionable to people (and no, not just to geographical experts).

…then maybe what is published would be more in line with what people prefer.

(Comparisons made herein are between NGS Winkel and Eckert III, unless otherwise specified)

How does NGS Winkel accomplish the big-tropics component of Winkel’s lower BARFF (compared to that of Eckert)?

Quoting daan again:

Notice that [on NGS Winkel] Africa and South America take up more of the map than the Eckert III map’s versions do, but less of the map than Eckert IV does. Notice that, in order to pay for this improvement in proportion, Winkel stretches Africa and South America out….

Yes.

Based on what daan said, in those quotes above, maybe the following paragraph could be given as the justification of NGS Winkel that could be given to potential purchasers of National Geographic magazine :

“We’ve, worsened shapes over most of the world, to make the non-arctic part bigger (in particular, we’ve vertically distorted Africa and South America to make them look bigger) so that you’ll think that (arctic area + antarctic area)/(tropical area) is ‘only’ 2.51 times as big as it really is, instead of 3.53 times as big as it really is.”

Oh thanks a lot :^)

So that’s the answer that was given to my question about NGS Winkel’s redeeming points and justification. Again, I thank daan for his answer. I agree with those answers.

I leave it to our audience to decide for themselves if they like what NGS Winkel gives (more, for instance, than they like what Eckert III or Apianus II gives).

Poll respondents have been agreeing, by a roughly 3:2 ratio, that Eckert III looks better, and shows the continents more globe-like and realistic, than NGS Winkel does.

daan has kindly provided posted images of Eckert III and NGS Winkel, so that, based on daan’s above-quoted answer, and on the images themselves, anyone can judge, for themselves, the relative desirability of Eckert III and NGS Winkel. (as did the poll-respondents, based on images of the two maps)

Now, let me reply to a few more comments in daan’s post.

Regarding Robinson, it would be nice to disclose a map’s projection. It would be nice for a “map” to be a specified mapping of lat,lon to X,Y.

Straw man. This was never in dispute.

Good. Then you agree that it would be nice if atlas publishers would specify the method by which they interpolate Robinbson’s tables. So why not tell them so.

Your thesis was that Robinson could not even be called a map.

_That_ is a straw-man. I’ve already retracted and qualified that statement, in a previous post here.

2. Regarding NGS Winkel, it would be nice for representations to look like what they’re supposed to represent.

Regarding any projection, it would be nice… but impossible.

I thought that it was well-agreed here that no map projection can accurately show the Earth’s surface, because the Earth is round and the map is flat.

Nevertheless, some maps manage to look a lot more ridiculously distorted than others. I discussed that above in this post—the matter of what is most noticeable and objectionable to people, and the matter of which map is shape-distorted over a larger % of the Earth. You say it’s just my own opinion. But evidently not, based on polling results.

I’ve been polling general audiences. …people who don’t have a prior opinion.

Notice how much of the [Eckert IV] map is taken up by Africa and South America. Referring to a globe or maps of those continents specifically, notice how ridiculously stretched Africa and South America are.

…but the difference is that at least Eckert IV, like Peters, gains something thereby—the equal-area property.

daan’s comments about area and shape are answered above, but I might as well directly answer a few comments here:

But most of all, look how ridiculously puny Africa and South America are compared to the map as a whole.

A number of answers to that:

For one thing, I must admit that I don’t notice it. According to daan’s figures, quoted earlier in this post, NGS Winkel shows the tropics a whole 13.7% bigger than Eckert III does :^) in comparison to the Earth’s area.

Of course the under-representation of area would be greater in the tropics as a whole than in continents that also extend out of the topics.

For another thing, Eckert III gives regions, even at the equator, their fair share of the overall longitude-extent of the map. It gives them their fair share of the east-west linear size. And Eckert III, being conformal at the equator, also gives them the north-south size that is right for that east-west size.

So, contrary to what daan wants to imply, Eckert III in no way shorts Ecuador. …or Africa or South-America, in regards to their portrayed size. Yes it magnifies the area at higher latitudes. NGS Winkel does that too. But it would be incorrect to say that that means that Ecuador, Africa, or South America are being shown incorrectly or unfairly small, as the abovequoted passage tries to imply.

As daan has earlier agreed, the Arctic’s magnification, in Eckert III, doesn’t come at the expense of Ecuador’s size or shape.

Now, each of these maps is ridiculous. None of them looks like reality. However, each of them has its place, and where one map excels, another fails, and there is no free lunch to be had. RogerOwens wishes you to subscribe to his belief that Eckert III is less ridiculous than Winkel tripel. I say piffle.

Poll-respondents say otherwise.

I mean…look at NGS Winkel :^)

Because both Eckert III and NGS Winkel magnify the Arctic, and both of them give an incorrectly high value for (arctic area+antarctic area)/(tropical area), then I emphasize that the matter of Eckert III vs NGS Winkel DOESN’T depend on discounting the importance of areal fidelity.

The matter of the importance of area, vs the importance of shape, is a different topic. In these posts I’ve been discussing two separate topics: Eckert III vs NGS Winkel, and area vs shape.

Even without discounting the importance of area (compared to shape), the fact remains that Eckert III and NGS Winkel both significantly magnify the Arctic, and the quantity
(arctic area+antarctic area)/(tropical area). …and NGS Winkel gives ridiculous shapes over a larger part of the Earth.

For certain practical purposes, sometimes equal-area is important, and sometimes conformality is important. Under those circumstances, one should make the appropriate choice between those two properties, and not use a “compromise” map that has neither property.

Don’t navigate by Hammer or Eckert III. Don’t use Winkel or Lagrange for a map intended to show what percentage of the Earth has what vegetation.

But we’re talking about circumstances when neither of those two properties is needed.

But yes, I do also claim that areal fidelity is aesthetically less important than good shape. Of course, partly that‘s a subjective impression, but I’ve been saying that the claim also has some objective justification. There are objective facts that can guide that subjective comparison of the aesthetic importance of shape and area.

More about that below.

I’m not saying that the arctic magnification of Eckert III and NGS Winkel is intentional. I’m saying that it’s a good thing for anyone wanting to closely examine the Arctic, and that it isn’t a bad thing for anyone—except for someone who needs an equal-area map for some specific purposes.

And I could just as well argue that Africa being stretched isn’t a bad thing for anyone—except for someone who needs shapes to be correct for some specific purpose.

You could, but there’s no particular reason why uniform area-scale should be as aesthetically important as shape. As I said, people expect representations to look like (i.e. be shaped like) what they’re supposed to represent. Different areal magnifications might be important to you as an individual, but there’s no particular reason to say that it’s inherently as aesthetically important as shape.

Certainly that directional scale-disparity would be bad for navigation. …for determining distances and directions.

But, we’re talking about circumstances where we don’t need conformality or equal-area, and are instead more concerned with aesthetics.

You say that areal-infidelilty, too, is a distortion, and you speak of balancing both kinds of distortion, of area and shape. You speak of them as if they were just two kinds of the same thing. You speak of them as if they were commensurable. You imply that it’s biased, prejudiced, and unfair when I aesthetically value shape more than area-proportionality.

On the contrary, I say that it’s biased and prejudiced to assume that they’re commensurable, two kinds of the same thing, and to be valued equally. All assumptions are subject to question.

People aesthetically like portrayals of things to look like what they’re supposed to portray. Whether and how much those portrayals are differently magnified has little if any effect on that.
(…but it could count separately, as a separate consideration to those who particularly notice it)

Yes, sure, some people might have an aesthetic aversion to _any_ departure from how the globe shows things. That would be a perfectly valid aesthetic evaluation.

But shape and areal-fidelity are two completely separate, different, and incommensurable matters.

So there’s no particular reason to _assume_ that both kinds of departures from the globe are equally unaesthetic.

People expect an image to be shaped like what it’s supposed to represent.

…and that’s less demanding than saying, “I want the map to be just like the globe’s surface. I want globe-fidelity in every regard, because, to me, that’s an aesthetic standard in itself.”

But even if you don’t share that exact-globe-surface-copy standard, a representation still looks better if it’s shaped like what it’s supposed to represent. That’s my point. That’s what makes shape different from areal-proportion.


Maybe, as one individual’s aesthetic preference, you’d also like the size-relation of two objects’ representations to be the same as it is for the two objects themselves. I suggest that, if it’s accepted that the map isn’t one big exact copy of the Earth’s surface, then there’s no reason to expect there to be that expectation or need for area-proportions to match those on the globe.

I can’t prove that you aren’t someone for whom aesthetics requires the map surface to be literally the same as the globe surface. But, if you aren’t, and some people aren’t, then the expectation for things to be shaped like what they represent is more basic and likely than an expectation for different places to be shown with the right mutual areal relation.

Anyway, by about a 3 to 2 ratio, poll-respondents agree with me that Eckert looks better and more realistic than Winkel.

, and they’re [Eckert’s shapes] even more of a mess because the very shapes that look good don’t fill up nearly as much of the map as they ought to

NGS Winkel shows the Tropics a whole 13.7% larger than Eckert III does :^)

(for a map of given area)

No, I haven’t measured Eckert’s and Winkel’s areas of Africa and South America. But we’ve determined their tropical areas, and that can stand in for Africa and South America. Of course, extending out of the tropics a bit, Africa, and especially South America, won’t be under-sized as much as the tropics are.

, leaving the global proportions misshapen and ridiculous.

“Misshapen” generally refers to shapes.

Both projections significantly magnify the Arctic and over-represent the quantity (arctic area+antarctic area)/(tropical area), as discussed elsewhere.

Winkel, but not Eckert, magnifies a latitude-band’s outer regions, in comparison to its regions near the central meridian.

Eckert III? I admit that Eckert III doesn’t portray Antarctica (1/36 of the Earth) as well as Winkel. But 1/36 isn’t the same as ¾, is it.

Well great; let’s just use Mercator! No matter that it portrays Antarctica as infinite in extent; Antarctica is only 1/36th!

…except that that doesn’t resemble what I said. Just before the passage you quoted, I said that I was answering the argument that NGS Winkel portray’s Antarctica’s _shape_ better than Eckert III does. So I was saying that accurate shape portrayal isn’t as important for 1/36 of the Earth as it is for ¾ of the Earth.

At no time did I say that the smallness of a region makes it ok to magnify it a lot.

But, as described above, with Eckert III, you don’t uglify and mis-represent how the continents look as much as Winkel,

This is just wrong. Eckert III mangles practically all of Canada, northern Europe, and northern Asia worse than Winkel tripel.

Yes, and those latitudes, and on up, comprise about ¼ of the Earth’s surface.

You go on and on about east-west compression on Winkel tripel, but those regions are stretched east-west on Eckert III. It’s fairly mysterious how you do not see that

I don’t deny it. It’s answered above in this post.

, but regardless, it’s right there in the distortion diagrams, and certainly I can see it quite obviously—just as I can see the markedly worse proportions in Eckert III. While average angular distortion for the two maps is similar across those regions

I answered that claim above in this post.

Anyway, look at NGS Winkel. :^)

By a 3 to 2 ratio, poll-respondents agree with me about the appearance and accuracy comparison of Eckert and Winkel.

“I did not write “more than a quarter more [arctic magnification for Eckert]”! With this sort of misquoting going on and on and on, the conversation cannot progress.”

I certainly apologize for the unintentional misquote. It was an isolated and accidental misquote, and not part of a consistent pattern of misquotes.

Yes, I feel that the reason why no one objects to NGS Winkel, in National Georgraphic magazine and other NGS publications, is simply because no one notices or cares

…and the ones who do notice and care disagree with you that it’s a poor choice.

Well, by a 3 to 2 ratio, the people responding to my polls say that Eckert III looks better and more accurately and globe-like portrays the continents.

Over most of the Earth, Eckert III’s shapes of continents and countries is more accurate than that of NGS Winkel. How’s that for education?

It’s horrible for education because it’s factually wrong

Incorrect. See above.

and because it ignores the other half of the equation: proportions across the map.

I don’t ignore that. I admit that the quantity (arctic area+antarctic area)/(tropical area) is misrepresented by a factor of 3.53 on Eckert III, and “only” by a factor of 2.51 on NGS Winkel

:^)

And don’t forget Winkel’s LAFF—a form of areal infidelity (defined above) not possessed by Eckert or Apianus II.

Anyway, I don’t agree with your belief that shape-distortion and areal infidelity are commensurable comparable quantities at all, to be similarly-valued—as discussed above.

But I repeat that that argument is in no way necessary for my criticism of NGS Winkel, because Winkel and Eckert both significantly over-represent (arctic area+antarctic area)/(tropical area), while Winkel looks worse than Eckert, shapes-wise, over most of the Earth (as discussed above).

One more thing: Speaking of world maps that could replace Winkel and Robinson, I’d like to add Lagrange and August to that list.

But, you’d agree with me on that only if you agreed with me about the overall matter of shape versus area. That’s why I’m mentioning them last.

I haven’t polled people about Langrange and August vs Winkel and Robinson. But I will.

A subsequent poll that I’ll do will show about 10 world map projections, including the ones that I’ve mentioned here, including Lagrange and August.

I’ll start that poll, at various websites, after my current Eckert vs Winkel poll has run sufficiently long .

Michael Ossipoff

[RogerOwens]

Typo:

I meant to say that Quartic uses Hammer's _central-meridian_ parallel-spacing. (...which, of course they both get from Azimuthal Equal-Area's central-meridian parallel-spacing)

Also: daan, I didn't mean to imply that you think that only flat-polar compromises are permissible, or that Lagrange or August are impermissible. I understand that you don't have any such prejudices or exclusions.

When I spoke of Lagrange and August, I just meant that _anyone_ wouldn't agree with me about approving them for an atlas, unless they're accepting of varying magnification, as am I.

I didn't mean to imply that you favored flat-polar maps. I mentioned their infinite e-w scale at the poles because I was saying that it seems inconsistent to tolerate that, but not a little extra variable magniication.

Anyway, looking at my posting, I noticed that it could seem as if I hadn't heard anything you'd said about disowning biases, exclusions and intolerance of variety. So I'm clarifying that nothing of the sort was intended.

Michael Ossipoff

[RogerOwens]

In my poll, some people who voted for Winkel told why: Its Antarctica is better, and they liked its realistically converging meridians. Those people like overall global pictorial realism, and that’s a valid preference. I’ll return to that subject below. Of course one requirement of global pictorial realism is that areas mustn’t be too far off, and so a conformal projection probably wouln’t do, for that purpose. But Aitoff, Hammer, and probably Quartic would be fine. Those who voted for Winkel would probably prefer those 3 maps to NGS Winkel.

As I mention below, global pictorial realism is valid artistically and decoratively, and seems best for kids’ first world map.

So it certainly has its place.

I’d like to clarify what I mean about considerations regarding shape vs area.

1. Reference
…a) Area
…b) Position/location, direction, distance

2. Education

3. Artistic/Decorative

4. Atlas world maps


First, as you were saying, for all 4 purposes, variety is desirable, and that’s the best answer. Equal-area maps, and conformal maps.

1a) Area Reference:

Usually, when I refer to maps, including world maps, it isn’t to find out relative areas. But sometimes it is, and when it is, I use an equal-area map. None other would make any sense for that purpose.

1b) position/location, direction, distance reference:

Position/location is a lot easier when shape is accurate. Likewise direction. Distance is a lot easier if, at any point, the scale is the same in every direction.

All of the considerations in the paragraph before this one recommend conformality for position/location, direction, or distance reference. …conformal maps such as Lagrange and August.

If areal-relation is referred to less often than position/location, direction and distance, that suggests Lagrange or August as the best choice.

But, on a wall in a home or classroom, or in an atlas, _both_ conformal and equal-area maps would be good if one wishes to have maps for both kinds of reference.

“Compromise” maps have no place for areal reference, and aren’t as good as Lagrange and August for position/location, direction or distance reference.

Do “compromise” world-maps really have any place at all, or are they just a fashion?

2. Ecucation:

I suggest that the first-introduced classroom-map, for the beginning school-grades, should be elliptical, because that best show’s the Earth’s round shape. That’s a worthwhile educational property for a first classroom map.

Of course a map for that purpose should be a good global pictorial representation of the Earth, and so areas should be good—or at least not too bad.

Apianus II is good for that purpose because of its great simplicity and its linearity. I think there’s merit to a projection whose construction is easily explained to the earliest school grades.

But its Antarctica is so distorted that it would be seriously misleading as an early-grades educational map. In fact its menacing octopus appearance could even cause nightmares. (It looks worst when the U.S. is at the central meridian, because of the longitude of Antarctica’s greatest radius.). So maybe if Apianus II is used for kids’ first world-map, the Antarctic should be left out. There could be a separare map of Antarctica.

Additionally, Apianus II has an artificiality about it, with its linear grid artificially put in an ellipse—The artificiality shows.

So, alternatively, Aitoff, probably the best-looking global-pictorial representation of the Earth, could be used instead. Its construction principle isn’t much more difficult to explain, even to early school-grades.

Though Hammer brings exact equal-area, that property isn’t needed by those early-grade students; and Aitoff looks better. The shear and centerward sweep of off-center continents is less likely to be enough to be perceived as objectionable.

So, Apianus II or Aitoff for first classroom map. Likewise for world map in a first atlas for kids.

For later grades, demonstrating the Earth’s roundness seems unnecessary, but area-reference could be needed, and so Hammer or Quartic would be desirable. But a good look at all regions, for reference to position/location, direction and distance in regions, is likely to be needed more often than area-reference, suggesting the desirability of Lagrange or August.

Of course August is better than Lagrange because less of its areal-magnification is in polar regions, and its max/min scale variation is less. And it’s conformal everywhere, even at the poles.

But, of course, again, having both views of Earth (conformal and equal-area) would be desirable.

Compromise maps? What for? To combine both kinds of failure?

Regarding August’s “split”, people might object, “The Earth isn’t split like that!” So point out to them that Lagrange (and most all world maps) split the Earth even more than August at the map’s poles. They split it so much that strips of surface that were adjacent to eachother are split from eachother by 180 degrees. So August is _less_ split than Lagrange is.

3. Artistic/Decorative:

First, I want to admit that surrealism is a valid kind of art, and that NGS Winkel is justified as surrealism, for those who want that.

But I’m more interested in more representational maps

Because of the flat-map, round Earth difference, good global and regional representational-ness aren’t compatible.

There’s no wrong/right regarding the choice between good portrayal of regions, vs a good overall global pictorial view of the Earth. It’s purely a matter of personal preference. I think that artistic or decorative variety calls for both kinds of maps.

As mentioned above, good overall global pictorial representation requires good areas—preferably exact or nearly-exact. …which disqualifies conformal maps. I’ve been suggesting Aitoff, Hammer, and Quartic for that purpose.

Lagrange and August are better at showing a good view of regions.

For the bests overall global pictorial view, surely Aitoff is the winner. …with Hammer close behind.

For a slight loss of realism, but with attractive stylized appearance, Quartic provides pseudocylindrical-ness.

Hammer and Quartic combine areal reference with the educational and artistic/decorative global pictorial merit.

Lagrange and August combine position/location, direction and distance reference with good regional appearance, giving them their own kind of equally-good artistic and decorative value.

Any one or more of Aitoff, Hammer, Quartic, Lagrange, or August would be good decoratively and artistically.

Ideally, for posting on a wall, it would be good to post both kinds of maps (equal-area and conformal). If it were desired to post only one kind, then it would depend on one’s artistic/decorative taste, &/or on the matter of which kind of reference is more often needed.

4. Atlas world map:

Often atlas world maps aren’t needed for regional reference, because atlases have larger-scale maps for that purpose. But sometimes one _does_ use an atlas’s world map for regional reference, because 1) it might be quicker than finding the right regional map; Or 2) maybe the region of interest is divided between 2 or more regional maps.

So there’s merit in an atlas’s world map accurately showing regions. But it might sometimes be needed for areal comparison’s too. So it’s difficult to say whether it should be Lagrange, August, Hammer or Quartic.

So why not use 1 or 2 extra pages to show _both_ kinds of map. …an equal-area world map for areal comparisons, and, additionally, a conformal one for regional reference or location/position, direction or distance.

Pages aren’t so scarce in atlases that it wouldn’t be feasible to include 2 world maps. Look at the many atlas-pages that are used for articles of all kinds.

As I was saying before, “compromise” maps are currently in vogue, but they obviously aren’t any good for either kind of reference, and don’t have either kind of aesthetic merit. Double failure isn’t merit..

As for the merits-comparison between different compromise-maps, I said what I wanted to about that in previous posts.

When I do a poll between about 10 or 15 world map projections, I expect that the people who voted for Winkel (because the better Antarctica and more convergent meridians give better global pictorial representation (in spite of more bad local shape)) will instead vote for more representational maps such as Aitoff, Hammer and Quartic. I expect that NGS Winkel won’t be anywhere near the top of the “finishing order” in the poll-results.

Michael Ossipoff

[RogerOwens]

Stereographic, interrupted on 2 opposite meridians:
----------------------------------------------------------------

Oops! I forgot to include the all-earth-showing conformal world map-projection that has been used more times than any other:

Steregraphic, interrupted on two opposite meridians, equatorial-aspect.

Interrupted on wo mutually-oppossite meridians. It's two side-by-side circular ,equatorial-aspect, Stereographic maps.

When a conformal world map is interrupted on only one meridian (as are most world maps), area magnifications are tremendous in some regions. I'm not saying that's bad. It's fine with me. But the old and popular Stereographic interrupted on 2 meridians, equatorial-aspect, has much less magnification of some regions over others.

So, in case some don't like large regional magnifications (such as large polar magnifications), it's still possible to have a conformal world map: Stereographic, interrupted on 2 opposite meridians.

By the way, of course the two circles needn't be side-by-side, though they always or nearly-always are shown that way.

They could be verticallly-arranged if desired, or if space-availability dictates it. Or they could be diagonally-oriented with respect to eachother.

Michael Ossipoff

[RogerOwens]

Lest I seem to speak too negatively about certain map projections, I want to emphasize that I don't greatly dislike any map, and criticism about maps is intended good-naturedly. The less-desirable map projections aren't so bad as to be a subject of major complaint. I clarified earlier that my comments aren't intended as social protest, but only as friendly criticism.

Robinson:

Obviously, with the Robinson projection, there was conscientious, and expertly successful, effort to minimize, mitigate, and balance the problems of size and shape.

Though I don't believe in "compromise projections", and don't agree that there's any need for them, I must admit that the Robinson projection does an excellent job of what it's intended for.

If only the Robinson projection were specified (by someone divulging what interpolation method is used).

Compromise maps don't tell you a whole heck of a lot. They're the HoHo s and DingDongs of the map-projection world.

Winkel-Tripel:

Something good can be said about every map projection.

NGS Winkel-Tripel can be justified on grounds of Euro-chauvinism, surrealism, and retro-nostalgia for the maps of the 16th century, that great age of discovery during which map-makers had access to only limited information about what was out there in the world.

/Variety:

Dan was right, to emphasize the desirability of variety in map projections. That's one reason why it's regrettable that, in atlases, and in National Geographic, all you'll find are two flat-polar compromise-maps. There are other kinds of maps. Some of them have properties.

Michael Ossipoff