Two polls on world map projections

[RogerOwens]

My comments last night were abrupt, spur-of-the-moment comments. So I'd like to add a few things:

Candidate Line-Ups:

Werner's candidatre line-up, of 8 projections, didn't contain any elliptical maps. Mercator was the only conformal map in the line-up.

The poll in which daan participated, with 9 projections, didn't include Hammer, a very popular, relatively recently popular, world map.

Neither poll included any conformal projection other than Mercator.

Those failings are difficult to justify.

Neither poll includes Eckert III. Eckert's maps aren't so widely-used anymore, and so I'm not saying that the poll-conductors should have known to include Eckert IILI. But, because Eckert III very consistently beat NGS Winkel-Tripel, in my 2-projection poll, now with 188 voters.

Actually, the matter of whether or not daan should have included Eckert III in his poll depends on wheither or not his participation in that poll took place after I announced the results of my Eckert III vs NGS Winkel-Tripel poll.

Balloting and Count:

In daan's post, there was no mention of what kind of voting was used in the polls that he reported, other than the poll in which he participated.

He mentioned that his poll used Round-Robin balloting: Each possible pair of candidates (maps) was presented to the voters, for them to vote between those 2 maps.

But, with 9 maps in the poll, that would mean that there were 36 pairs between which the voters had to vote.

That's asking a lot of the voters. Most likely, most of the voters didn't vote between all of the 36 pairs of projections. That brings into question the meaningfulness of the results.

A much better way to conduct a many-alternatives poll: Do rank-balloting. Collect a ranking from each voter, and then do a pairwise-count that gives a Round-Robin result.

There are many pairwise-count methods. Each of them would elect a candidate that pairwise-beats each of the others. That candidate is called the "Condorcet Winner".

But there's the question of what to do when there's a cycle, where, for example, A beats B beats C beats A. ("beats" meanss "pairwise-beats").

When I reported my 8-candidate poll, I also reported the balloting and the count-rule that I used. (I used the popular Maximize Affirmed Majorities (MAM) method).

daan didn't tell us how his finishing-order, his results-ranking, was gotten. Maybe his found the Condorcet winner (reporting it in 1st place), then, removing it from the count, determined the Condorcet winner again (reporting that winner in 2nd place), ...etc. That's the usual way to produce an output-ranking.

But daan didn't report what he did, in his count, when there was a top-cycle instead of a Condorcet winner.

Though, in polls, a top-cycle for 1st place is very rare, cycles are common for other positions in a poll's output-ranking.

The top positions of NGS Winkel-Tripel and Robinson in the output rankings of the polls that daan reported lose their meaning, given the above-listed objections to the candidate-lineup, balloting, and incomplete reporting for those polls.

...in particular, the absence of Hammer and Eckert III in the candidate-lineups.

Michael Ossipoff

[daan]

Those failings are difficult to justify.

Not, and they’re not failings. You ignored the abstract, which you have free access to and which explains the purpose. The purpose was not to discern some magical “most favorite”, particularly not out of an unbounded space of projections. Different demographics have different preferences. The study explores the differences between professional and lay, male and female, and, it turns out, between Americans and East Indians, all within a sample of common projections, with no pretense of some set representative of any arbitrary qualities some random person might come up with. It further explores the preference for rounded corners versus angular, elliptical versus sinusoidal meridians, straight versus curved parallels, and pole lines versus points. None of this was to determine some mythical favorite, but to confirm that there are in fact differences and preferences for traits. Please read the abstract again in order to see what the purpose of the study was.

That's asking a lot of the voters. Most likely, most of the voters didn't vote between all of the 36 pairs of projections. That brings into question the meaningfulness of the results.

A much better way to conduct a many-alternatives poll: Do rank-balloting.

No, any trial that didn’t make it all the through was discarded. This was a relatively large-scale study. There was no need for any of the tie- or cycle-breaking convolutions. The study’s method is much better than rank-balloting.

But daan didn't report what he did, in his count, when there was a top-cycle instead of a Condorcet winner.

The study reports everything, of course. I understand visiting a library can be inconvenient.

The top positions of NGS Winkel-Tripel and Robinson in the output rankings of the polls that daan reported lose their meaning, given the above-listed objections to the candidate-lineup, balloting, and incomplete reporting for those polls.
what he did, in his count, when there was a top-cycle instead of a Condorcet winner.

Not to be rude, but no, they don’t lose anything, and most of your speculations are meaningless because you’re speculating on your own speculations. Refer to the original study, please.

Regards,
— daan

[Atarimaster]

Hello daan,

after you’ve mentioned the User preferences for world map projections paper again in the Eckert IV/VI-thread, I’ve finally read it.
Very interesting piece of work!

But to be honest, I don’t quite get the choice of some pairings within the study.
The first thing that made me frown was the Wagner IV vs. McBryde-Thomas Flat-Polar Sinusoidal pair on the "curvature of meridian lines" test. My first thought was that Wagner IV vs. Wagner I or Eckert IV vs. Eckert VI might have been a more appropriate pair. But then, I thought you (with "you" being plural here, meaning Mr. Jenny, Mr. Savric and yourself) decided to use the McBryde-Thomas projection because the "sinusoidal nature" of the meridians is more prominent in this projection.
Okay, problem solved.

But then, on the "curvature of parallel lines" test, there are the pairs Robinson vs. Wagner VII and Natural Earth vs. Wagner VII with rounded corners.
In both cases, the curved parallels projection is equal-area, while the straight parallels projection is not. In my opinion, this contradicts the study’s postulate to test pairs of graticules with "very similar, if not identical, characteristics – except for one varied, tested characteristic" and includes the danger that the straight parallels projection is chosen not because of the kind of parallels but because the participant prefers compromise over equal-area projections (i.e. the looks of them of course, since the lays might not even be aware that there are different metric properties).

My choice would have been Eckert IV vs. Wagner VII with rounded corners resp. Wagner IV vs. Wagner VII.
Or maybe, just in case you’d prefer not to run this test with equal-area projections only, replacing the latter with Wagner VI vs. Wagner IX (although that might have been a bad idea since it introduces two additional projections to the lineup).

Don’t get me wrong, I’m not trying to say that I’m smarter than you guys. Maybe I’m just failing to see what’s wrong with the pairs that I’ve mentioned… So, I’m just trying to understand your choice.
…
…
Okay, well, since I really love curved parallels and don’t want accept that the participants strongly preferred straight parallels, I’m probably a bit like the guy who is blaming the referee when his favorite soccer team screwed up again.

Kind regards,
and happy holidays,
Tobias

[daan]

« laugh » I had to read your posting twice to make sure I wasn’t the one who wrote it. We do seem to view the projection world very similarly.

But then, on the "curvature of parallel lines" test, there are the pairs Robinson vs. Wagner VII and Natural Earth vs. Wagner VII with rounded corners. In both cases, the curved parallels projection is equal-area, while the straight parallels projection is not. In my opinion, this contradicts the study’s postulate to test pairs of graticules with "very similar, if not identical, characteristics…

I brought up this identical objection in a session at the 2014 AutoCarto in Pittsburgh. The first authors (Šavrič, Jenny) delivered a presentation there before completing the paper. Later I was invited to co-author because I had provided mathematical assistance and important technique. Honestly I do not remember how Bojan Šavrič answered my objection, but I thought his response was about as reasonable as it could have been under the circumstances. I agree this factor is a weakness in the study.

Like you, I generally prefer curved parallels. They offer the potential to reduce angular deformation out toward the edges, without, in my opinion, burdensome side effects. If the map displays no graticule, it probably should keep straight parallels, but otherwise, why would you •not• use curved?

Happy holidays to you as well.
— daan

[RogerOwens]

daan Strebe said:

If the map displays no graticule, it probably should keep straight parallels, but otherwise, why would you •not• use curved?

Because:

1. For climate and vegetation data-maps, easily-comparable latitude is useful and valuable, and is much easier if parallels are actually parallel (which of course is why they're called "parallels").

2. In fact, latitude is always a factor (even though not the only one) in temperature and other climate-considerations, which are often of interest, even if the map isn't a data-map. So ease of comparison of latitudes is helpful in general.

3. On a cylindroid (cylindrical or pseudocylindrical) map, it's relatively easy to find Y from Lat, or Lat from Y.

...but not with curved parallels.

Michael Ossipoff

[daan]

daan Strebe said:

If the map displays no graticule, it probably should keep straight parallels, but otherwise, why would you •not• use curved?

Because:

1. For climate and vegetation data-maps, easily-comparable latitude is useful and valuable, and is much easier if parallels are actually parallel (which of course is why they're called "parallels").

We’ll just disagree here. The point of the graticule is to tell you where you are on the map. I rather doubt most people find it hard to figure out where a location is just because parallels curve moderately. We have bazillions of maps with bowed meridians, but meridians don’t bow. The fact that parallels are called “parallels” does not suggest people get confused when they’re not displayed parallel.

2. In fact, latitude is always a factor (even though not the only one) in temperature and other climate-considerations, which are often of interest, even if the map isn't a data-map. So ease of comparison of latitudes is helpful in general.

This is the same point as (1), with the same refutation. Just look at the graticule. That’s what it’s there for.

3. On a cylindroid (cylindrical or pseudocylindrical) map, it's relatively easy to find Y from Lat, or Lat from Y.

...but not with curved parallels.

Hogwash. Plenty of pseudocylindric projections require iterative solutions for y from φ, and plenty of projections with curved parallels have straightforward, non-iterative generating functions.

— daan

[RogerOwens]

Reply to daan's earliest un-replied-to posting:

Re: Eckert IV vs Eckert VI. Other related topics.
by daan » Sun Jan 10, 2016 2:52 am


I’d said:

Eckert III, like Apianus II, could be vertically-expanded to move its conformality up to a point at lat 45, lon 0. I’d probably prefer that.

You reply:

That would wreck your beloved Africa’s shape, much worse so than Winkel tripel.

It would make Africa skinny. More than Winkel-Tripel? On equal-area Eckert IV, sure. But on Eckert III? I don't know. Maybe-so.

All of the equal-area high-space-efficiency maps that I’ve looked at, even the ones with a point-pole, seem to have a skinny Africa. But of course especially the equal-area ones.

I guess high space-efficiency is incompatible with good shape-proportions everywhere along the central-meridian, especially with equal-area.

But, though Africa’s shape is undeniably what gives us our first impression of the map’s shapes, it’s also true that good shapes at lat 45 would balance the distortion of shape-proportions in the arctic and tropics.

And, even in Florida, where I reside, I’m considerably closer to lat 45 than to lat 0. Probably most of the world’s population is closer to lat 45 than to lat 0.

Now I’d choose conformality at (lon 0, lat 45), for a pseudocylindrical, or along the 45th parallel, for a cylindrical. (Note that a cylindrical has the advantage of conformality along parallels of the mapmaker’s choice, rather than only at 2 points.)

Besides, for PF8.32, the aspect ratio giving conformality at more poleward latitudes reduces the horizontal-ness of the outer meridians in the arctic.

Of course incompatibility between areas and shape-proportions is a price paid for high space-efficiency.

I’d said:

I’d say, I like all maps except Winkel Tripel.

It has unnecessarily abominable shapes over most of the Earth. It has no properties.

You reply:

It has no properties you care about or that you care to identify. It has “properties”. Any map has “properties”.

You’re right. I take it back: Winkel-Tripe has properties:

1. It’s flat (because it’s a flat map)
2. It’s interrupted on only one meridian.
3. Its border is a convex closed curve.
4. It has line-poles.
…etc

I didn’t mean that Winkel doesn’t have any attributes at all.

Here’s what I mean by map-merit-comparison properties:

1) Properties are Yes/No attributes. …attributes that answer “Yes” to a question that only has a Yes/No answer, and for which the “Yes” answer is the favorable one.
.
2)To be of any use for comparison, a property must not be an attribute of all flat world maps.

(What follows is a bit vague)

3) A property should refer directly and explicitly to something that directly benefits usefulness and/or accuracy. A property should answer favorably and affirmatively to a question that only has a Yes/No answer. If a property is to qualify by bringing another desirable attribute, then that other desirable attribute should answer favorably and affirmatively a question that only has a Yes/No answer.


4) Of course an attribute isn’t a useful comparison-property if it’s had by all the maps being compared in a particular discussion.


Winkel-attributes #1, above, fails criterion #2.

Winkel-attribute #2 fails criterion #4.

Winkel attribute #3 fails criterion #4, because Mercator isn’t being compared in this discussion of line-pole maps (…because Mercator doesn’t show the pole at all).

Winkel attribute #4 fails criterion #3, because, though attribute #2 brings some improvements, they aren’t Yes/No attributes.

Some examples of properties are: equal-area, conformality, linearity, equidistance (of parallels), cylindroidness, conformality along any +/- pair of parallels chosen by the mapmaker, uniform centimeters per degree of longitude throughout the map, parallels and meridians always crossing perpendicularly…etc.

Cylindroidness is a property that is really a combination of several properties:

Parallels are straight lines.
Parallels are parallel.
Each parallel has constant scale along it.

(I’m not saying that the following desirable attributes are properties)

Sinusoidal-like maps have good central-meridian shape-proportion and nearly equidistant parallels even when they’re equal-area. And are nearly equal-area even if their parallels are equidistant. But I don’t call that a property, because it isn’t definite enough. It could be called a near-compatibility of two properties on Sinusoidal-Like maps.



I’d said:

Yes, but, due to Robinson and Winkel-Tripel, how many kids think that the world is shaped like a chamber-pot?

You reply:

None?

Maybe, maybe not.

Does Mercator make kids think that Greenland is bigger than South America? Not if they’re told otherwise. And aren’t they always told otherwise? I can say that I was often told that.

All maps have distortions. Then does that mean that they always convince people that those distortions are Earth-attributes and not distotions?

With, at every point, the same scale in all directions, conformal maps’ distortion consists of distortions shared by all flat maps: Varying scale; most great-circle distances inaccurate; most great-circle directions inaccurate.


…and area-distortion, which, of course, is not shared by the equal-area maps.

I’d said:

Directions are relevant even if you aren't sailing. Accurate directions require conformality.

You reply:

Yes, as long as you mean purely local directions.

…and loxodrome directions, accurate everywhere only on Mercator.

Loxodromes are relevant even when not on water.

I’d said:

I have a Mercator on the wall. I consider it the most useful world map, for anything other than areas.

You reply:

…and anything to do with the poles…

…but Mercator can, and often used to, show all of Greenland, and even farther north than that.

How often do you really need to examine land closer to the pole than that?

Yes, nowadays, Mercator rarely if ever shows all of Greenland. But that isn’t because of a deficiency of Mercator. It’s the result of publishers feeling bullied by cartographers constantly criticizing Mercator’s magnified Greenland (as if every map didn’t have distortions).

In fact, Mercator is becoming difficult to find at all for that reason. My Mercator wall-map may soon be a valuable and unobtainable antique.

Speaking of the poles, your beloved Robinson and Winkel-Tripel distort the poles into lines, and have infinite scale along those lines.

, great circle routes, distances…

As we’ve discussed before, neither you nor I usually use a map to get great-circle distances. They’re more accurately gotten by calculating them.

And _all_ maps distort most great-circle distances. Yes, as a special custom choice, one can choose a map that shows accurate great-circle distances from one or two places.

Michael Ossipoff

[Atarimaster]

I don't have much time at the moment, so I'll restrict myself to two comments:

Does Mercator make kids think that Greenland is bigger than South America? Not if they’re told otherwise. And aren’t they always told otherwise? I can say that I was often told that.

Well, obviously nobody told Arno Peters.
He actually said in one of his presentations that he used to think that the Mercator map shows true areal relationships.
(At least, in case the transcription of that presentation given on this german website is trustworthy.)

The thing is, every teacher knows that the earth is (nearly) spherical, but do they know about the areal relationships? I really doubt that.

Yes, nowadays, Mercator rarely if ever shows all of Greenland. But that isn’t because of a deficiency of Mercator. It’s the result of publishers feeling bullied by cartographers constantly criticizing Mercator’s magnified Greenland

Sorry, but that's merely an assumption.
I don't have anything but an assumption, too, but mine's completely different:
I think publishers don't ever even listen to cartographers.

Eckert (1921) and Wagner (1949) (and probably other cartographers as well) both critisized the Mercator projection as being inappropriate for anything but navigation, but that didn't stop publishers from using Mercator as wallpaper map and in many atlases again and again. If they had listened to the cartographers, Peters most likely wouldn't have had a Mercator map in the mid-1960ies, and would've never started designing an own projection in the first place.

I think publishers just want to save space, and so they say: "Hey, let's cut off Greenland. Total waste of space, it's not as if anybody lives there or anything. At least nobody worth mentioning."
In my school atlas, all world maps – physical, politcal and even all of the (very large number of) thematic maps – were Winkel Tripel, with Greenland cut off at about 80° N and Antartica cut off completely. I can't imagine that this had something to do with Winkel Tripel's magnification of polar regions...

Regards,
Tobias

[daan]

I’d said:

Eckert III, like Apianus II, could be vertically-expanded to move its conformality up to a point at lat 45, lon 0. I’d probably prefer that.

You reply:

That would wreck your beloved Africa’s shape, much worse so than Winkel tripel.

It would make Africa skinny. More than Winkel-Tripel? On equal-area Eckert IV, sure. But on Eckert III? I don't know. Maybe-so.

I was talking about Eckert III, not IV, because you were talking about Eckert III, not IV. Convinced?

Top: Winkel tripel (Winkel formulation); Middle: Eckert III stretched so that [0°E, 45°N] is conformal; Bottom: angular error of middle map.

Eck45.jpg (300.65 KiB) Viewed 1945 times

But, though Africa’s shape is undeniably what gives us our first impression of the map’s shapes

Highly deniable, and vigorously denied by me in past postings. Your claim is unsupported by any research, any credible poll, and any powerful theory. What you say may have statistical support, but until that is demonstrated, you are merely spouting a highly deniable personal belief.

And, even in Florida, where I reside, I’m considerably closer to lat 45 than to lat 0. Probably most of the world’s population is closer to lat 45 than to lat 0.

You shouldn’t be rationalizing 45° by speculating that most of the population is closer to it than to 0°; you should go figure out what symmetric pair of latitudes lies closest to the most people and then propose that as a reasonable latitude at which to place the points of conformality.

I’d said:

I’d say, I like all maps except Winkel Tripel.

It has unnecessarily abominable shapes over most of the Earth. It has no properties.

You reply:

It has no properties you care about or that you care to identify. It has “properties”. Any map has “properties”.

I didn’t mean that Winkel doesn’t have any attributes at all.

Here’s what I mean by map-merit-comparison properties:

1) Properties are Yes/No attributes. …attributes that answer “Yes” to a question that only has a Yes/No answer, and for which the “Yes” answer is the favorable one.
…
Some examples of properties are: equal-area, conformality, linearity, equidistance (of parallels), cylindroidness, conformality along any +/- pair of parallels chosen by the mapmaker, uniform centimeters per degree of longitude throughout the map, parallels and meridians always crossing perpendicularly…etc.
…

Without getting into the tedium of rebutting much and pointing out the myriad “properties” you left out, I will state that metrics such as “minimum error” are also properties when defined rigorously, and of course you left out mention of every useful property that doesn’t suite your fixation on “cylindroid” maps in equatorial aspect. Your choice to ignore properties you don’t like doesn’t give you a convincing argument.

Speaking of the poles, your beloved Robinson and Winkel-Tripel distort the poles into lines, and have infinite scale along those lines.

Hilarious. After all I’ve written on the topic, and how I even embarked on my map projection avocation, you’d write something so ignorant about my preferences. And I really can’t grasp what your intent is in pointing out the deficiencies of the pole line when your beloved Eckert III and any of your beloved cylindrics all carry the same defect—except, of course, Mercator, which has got no pole to even stretch into a line.

, great circle routes, distances…

As we’ve discussed before, neither you nor I usually use a map to get great-circle distances. They’re more accurately gotten by calculating them.

As I keep having to point out, utility isn’t yes/no. Some maps distort distances less than others. Less results in better apprehension of… distances, which are a really rather important part of a space. And •any• property is more accurately gotten by calculating it—including whether or not the pole really is a line or a point. So if that’s your argument, then it really doesn’t matter what projection we use. Just do your calculations on the sphere, where they belong.

— daan

[RogerOwens]

Tobias—

True, I can’t say for sure that people are sufficiently warned about Mercator’s magnification.

Arno Peters’ experience was different from mine, and of course I can’t say which experience is more typical.

But Mercator should, and certainly could, always be accompanied by such a warning.

I agree that Mercator wouldn’t be a world map to present to early school-grades.

RogerOwens wrote:Yes, nowadays, Mercator rarely if ever shows all of Greenland. But that isn’t because of a deficiency of Mercator. It’s the result of publishers feeling bullied by cartographers constantly criticizing Mercator’s magnified Greenland

Sorry, but that's merely an assumption.

True enough, it’s an assumption. I don't know why publishers would choose to make that change now, and not before. Maybe because something has changed. ...something in the opinion-climate.

I don't have anything but an assumption, too, but mine's completely different:
I think publishers don't ever even listen to cartographers.

Eckert (1921) and Wagner (1949) (and probably other cartographers as well) both criticized the Mercator projection as being inappropriate for anything but navigation, but that didn't stop publishers from using Mercator as wallpaper map and in many atlases again and again. If they had listened to the cartographers, Peters most likely wouldn't have had a Mercator map in the mid-1960ies, and would've never started designing an own projection in the first place.

Yes, it’s odd that the objections were being made so long ago, and the replacement of Mercator is taking place so much later. One explanation could be that it just took criticism over a long time, to eventually influence publishers, store-owners, teachers, etc., and/or the people who buy the wall-maps or atlases.

But of course I certainly don’t claim to know the answer to that. Of course I can’t claim to understand and explain other people’s motivations.

I think publishers just want to save space, and so they say: "Hey, let's cut off Greenland. Total waste of space, it's not as if anybody lives there or anything. At least nobody worth mentioning."

Maybe, but there used to be Mercator wall-maps that showed all of Greenland. …suggesting that something has changed over time. …maybe for some identifiable reason, but maybe not. The question is, why did publishers eventually make that decision. I don’t claim to know. As you said, my suggestion was only an assumption.

In my school atlas, all world maps – physical, politcal and even all of the (very large number of) thematic maps – were Winkel Tripel, with Greenland cut off at about 80° N and Antarctica cut off completely. I can't imagine that this had something to do with Winkel Tripel's magnification of polar regions...

Agreed. It evidently was just (as you said) a perception that the Arctic and Antarctic didn’t matter.

But the question remains, why did publishers previously make Mercator showing all of Greenland, and now don’t do so? Of course just because there’s a question doesn’t mean that there’s an available answer.

Michael (“RogerOwen”)