How do publishers actually interpolate for Robinson?

[daan]

Yes, it's great that the Robinson mapmaker his version of the Robinson projection. But wouldn't it be kinda nice if he would share it with his public?… It has been pointed out (sorry, i don't have the citations) that those numerical scores are hardly worth the paper that they're written on. If a map has a high score, and looks iike crap to a lot of people, what does the score mean? Appearance and aesthetics aren't about numerical scores.

You cannot have it both ways. If you cannot tell the difference between two Robinson maps by looking at them, then the only significance to your complaint about not knowing the interpolation function would be for analytical purposes. But you have disclaimed the importance of analytics in the design. I am finished with the portion of the conversation; your arguments contradict each other and seem contrived merely to avoid backing down from an ad hoc accretion of preferences promoted as somehow more objective than they are.

I don't claim to use or encounter all map projections. But, with the exception of Robinson's, the specifics of all projections can be found, are available to the public.

Not so. Dietrich (Grundzüge der allgemeinen Wirtschaftsgeographie', 1927) published a thematic map on an unknown epicycloidal, evidently equal-area projection that Kitada (1958) derived an approximation for in absence of any published information. Bertin, famous for his monumental Semiology of Graphics (Sémiologie graphique) published maps on several projections that have not been described anywhere that I have found. Many 16th century maps have no known construction methods, but the graticules are evidently rigorous. There are others; Snyder worked out several himself from scattered periods whose construction methods were unclear. There’s also the question of whether a particular projection’s construction is casually available to “the public”. Until Maling in 1972, rather few were.

And no, it isn't just me. Several cargographers agree with me that Peters' projection looks really bad due to its distortion of those two continents for which shape distortion is particularly noticible--Africa and South America.

Firstly, I never argued in favor of Peters, so this is a straw man. You keep bringing up Peters as if Peters is the equal-area alternative. It‘s not, so enough with the Peters-baiting.

Secondly, the consideration is one of extremes, not absolutes. Gall–Peters is extreme. Robinson is not, and this is where you have another case of self-contradictory cherry-picking. It was Robinson who tendered the famous complaint about the Gall–Peters, “like wet long johns hanging on the line”. While he objected to the Peters portrayal, obviously he does not share your fanaticism for a precise-looking Africa. Robinson had a lot more considerations in mind than Africa when he created his projection. It just may be that his eye for geography was a lot more trained than yours, and the myriad distortions I keep pointing out that you keep claiming are not noticeable were in fact noticeable to him, to me, to map publishers, to geographers, and to many other people. It’s fine you have preferences. It’s not fine that you insist those preferences are shared by “most people” or that they have some greater objectivity or legitimacy than other preferences. They don’t. Until you come up with rigorous studies that show otherwise, you’re just fantasizing that what goes on inside your head should be more important to other people than what goes on inside their own heads.

When people say that a problem doesn't have an exact solution, they ususally mean that it doesn't have a solution that can be written in closed form… When someone tells you that a solution requires the use of numerical methods, they don't mean that you have to press the square root key.

Given that I have a formal education in numerical methods and thirty-five years’ experience in the topic, I’m not inclined to debate this stuff. You meant “closed-form”, not “exact”, it turns out. Definitions are not solipsistic. Whether or not something affects you does not change the generally agreed-upon definition. The person who implemented the programming behind that square root key used a numerical method to do it. “Closed-form” is not synonymous with “exact” and “numerical method” is not synonymous with “approximation”.

If you want a great-circle distance, or a loxodrome distance, then calculate it rather than measuring it on a map… Instead of stating a nomional scale, intended to be used for the whole map, it would be much better to state the distance, in miles, between the graticule parallels.

Sorry; I find that argument silly. North-south distances aren’t more special than other directions. And this is just another case of having it both ways. If distances are important, you say, then calculate them rather than measure them off a map. But now suddenly some distances—the special ones you like—are important to be able to measure off a map.

Better than Goode's Homolosine, or Interrupted Sinusoidal?

Interruption is yet another kind of distortion. Those gaps are distances. You can’t just pretend they don’t exist. Dealing with them implies knowing the great circle route.

If you want a route-distance, or if you want a quickly measured or estimated great-circle or loxodrome distance that uses only a nearby map, then use a conformal projection. Winkel Tripel isn't a conformal projection, and therefore would be a poor choice for measuring distances.

I have no idea what you mean by a “nearby map” but this argument is apples-to-oranges. Winkel tripel is near the best for a random selection of distances of random lengths. Conformal maps are useless for long distances.

Best,
— daan

[RogerOwens]

(Note: When I say “Winkel”, I’m referring to NGS’s vertically-expanded version of Winkel-Tripel.)

Yes, it's great that the Robinson mapmaker knows his version of the Robinson projection. But wouldn't it be kinda nice if he would share it with his public?… It has been pointed out (sorry, I don't have the citations) that those numerical scores are hardly worth the paper that they're written on. If a map has a high score, and looks like crap to a lot of people, what does the score mean? Appearance and aesthetics aren't about numerical scores.

You cannot have it both ways. If you cannot tell the difference between two Robinson maps by looking at them, then the only significance to your complaint about not knowing the interpolation function would be for analytical purposes.

Yes, it isn’t about the maps’ appearance, because, as you point out, the Robinson versions are identical.in appearance.

Then what is it about? It’s a suggestion that it would be good to state a map’s projection.

In part, you can consider it an aesthetic consideration. But it’s an aesthetic consideration that has nothing to do with map-appearance, and is unrelated to, and doesn’t contradict, my questioning of the worth of measuring good-appearance or merit by someone’s numerical score consisting of some weighted average of distortions.

But it isn’t only aesthetic: If it’s necessary to convert between lat/lon and X/Y on that published map, I’d prefer to do so in terms of the projection that was used to construct the map—silly me :^)

You say it might not make any practical difference? “Might not” isn’t good enough. I don’t want to have to take someone’s word for that.

But you have disclaimed the importance of analytics in the design.

Incorrect. I’ve disclaimed the meaningfulness of someone’s weighted-average numerical score as a measure of how good a map looks, or how good it is for the general public. As I said, appearance-aesthetics isn’t about numerical scores.

Below, you want to say that that somehow contradicts my wish that a map’s projection be disclosed, whether for aesthetic or practical reasons, as mentioned above.

[Y]our arguments contradict each other.

I commend you for that bold reach. Preferring a genuinely defined and stated map projection doesn’t contradict not recognizing the meaningfulness of someone’s arbitrary weighted average of some set of distortions as a measure of a map’s good appearance, usefulness, or merit.

and seem contrived merely to avoid backing down from an ad hoc accretion of preferences promoted as somehow more objective than they are.

“Ad hoc”? That means, “made up only for this particular purpose or instance” (such as evaluating Robinson or Winkel).

No, I’d object to any “projection” that isn’t specified, defined. Not just Robinson. And I regard any unnecessary, particularly blatant, shape-distortion as being really unappealing aesthetically-- in general, not just for Winkel.

So it isn’t entirely clear what you mean by “ad hoc”.

As for “objective”, you know that I’ve been repeatedly emphasizing that aesthetic preferences are a subjective individual matter. But my suggestion that projections should be defined _does_ have objective merit. Ugly shapes? That’s subjective. A distorted continent-shape that seems revolting to me, might be beautiful to you.

But yes, I’ll say now that I suggest that there _is_ some objective merit to my claims about shape-accuracy vs areal-fidelity.

1. I claim that a somewhat wrong shape is a lot more noticeable than a somewhat incorrect area-relationship (say, the amount of areal infidelity that would be caused by fixing the shape-distortion)..

2. I claim that it’s pointless and silly to make an issue about the areal relationship between continent, country or island representations that, due to their shape distortion, don’t resemble what they’re supposed to represent. …and, just about (in Winkel and Peters), are only identifiable by their labeling, and relative positions.

…unless you specifically want an area-cartogram, disregardful of shape as are all cartograms.

3. I claim that, even if noticed, areal-infidelity is neither ugly, anti-useful, nor objectionable. Map-users shouldn’t expect a flat world map to be free of all distortions. But grossly and unnecessarily distorted shapes are what’s ugly, not incorrect sizes.

4. Areal-infidelity needn’t be unfair, and needn’t be regarded as shrinking or taking area away area from the places not magnified. For example, in Eckert III, a region on the equator gets its fair share of the available X-width, because the X scale is constant on each parallel, including the equator. And, because the north-south scale at the equator is equal to the scale along the equator, that equatorial place also gets the amount of Y-height that is right for it. Thus, no one can rightly claim that an equatorial place is somehow cheated or shorted when Greenland is magnified. Yes, Greenland is magnified, but not by shrinking Ecuador, not at Ecuador’s expense.

Sure, there are special applications where equal-area is needed, and I’ve more than once made it clear that, when equal-area is needed, and when it requires shape-distortion, obviously that shape-distortion is justified. I gave Eckert IV as an example, and Mollweide. Even Gall-Peters, though I personally wouldn’t wall-display it.

In fact, I also said that Winkel’s vertical expansion, distortions and all, is justifiable for the goal of making the map bigger, to make better use of available page-space. …but that doesn’t mean we have to pretend that it looks good. As I said, I personally wouldn’t choose distortion-causing 1-dimensional expansion for making the map bigger, because I feel that the shape-distortion tends to make it harder to find things in the map, somewhat defeating the purpose of making the map bigger.

Likewise, I’ve acknowledged that Winkel’s better representation of Antarctica could justify its use for someone to whom that is important.

But there are equal-area maps, such as Hammer-Aitoff, whose shape-distortion is at the map’s periphery and doesn’t look bad, because it merely enhances the map’s realistic globular appearance. …and whose Antarctica is better, in size, shape and appearance, than that of Winkel.

I’ve said that what’s ugly to me might be beautiful to you. You’ve portrayed me as a connoisseur of Africas. What might such a connoisseur say, to perhaps explain Winkel’s actual emotional appeal to NGS?:

“For Africa, Winkel achieves a nice pointed effect, with distinct notes reminiscent of fine old maps of 15th and 16th century vintage, picturesquely distorted due to the usual unavailability of accurate longitudes during those classic vintage years.”

I don't claim to use or encounter all map projections. But, with the exception of Robinson's, the specifics of all projections can be found, are available to the public.

Not so. Dietrich (Grundzüge der allgemeinen Wirtschaftsgeographie', 1927) published a thematic map on an unknown epicycloidal, evidently equal-area projection that Kitada (1958) derived an approximation for in absence of any published information. Bertin, famous for his monumental Semiology of Graphics (Sémiologie graphique) published maps on several projections that have not been described anywhere that I have found. Many 16th century maps have no known construction methods…

Thankfully they aren’t among the maps that we frequently encounter.

And no, it isn't just me. Several cartographers agree with me that Peters' projection looks really bad due to its distortion of those two continents for which shape distortion is particularly noticible--Africa and South America.

Firstly, I never argued in favor of Peters, so this is a straw man. You keep bringing up Peters as if Peters is the equal-area alternative. It‘s not

Yes, I emphasize that t there are many other equal-area projections. But Peters is the natural, famous, spokesman and swimsuit-model for what can happen to shapes.

As you mentioned, Peters is the extreme example of blatant and prominent shape-distortion. Peters is the champion.

Admittedly, Winkel and Robinson aren’t as “FUBAR” as Peters. …but there’s no shame in coming in 2nd and 3rd.

But I probably implied that you were a Peters fan, and for that I apologize.

Secondly, the consideration is one of extremes, not absolutes. Gall–Peters is extreme. Robinson is not, and this is where you have another case of self-contradictory cherry-picking. It was Robinson who tendered the famous complaint about the Gall–Peters, “like wet long johns hanging on the line”. While he objected to the Peters portrayal, obviously he does not share your fanaticism for a precise-looking Africa.

You aren’t being entirely clear with us regarding what my cherry-picking contradiction is.

I thought that I clarified that it isn’t only about Africa. I criticize any and all blatant, prominent, and unnecessary shape-distortions, even when they aren’t of Africa.

But maybe it would be a good idea to add to, clarify, what I said before, about Africa and South America:

Africa and South America are continents, good-size landmasses. Big enough to be prominent and very noticeable. But small enough that it’s clear (by looking at a globe or a satellite photo) what their shape should look like. In contrast, for example, the Pacific Ocean, or even Eurasia, covers so much globe that it isn’t as easy to say when it’s shape is well or poorly portrayed on a flat map. A little shape distortion of the Pacific Ocean or Eurasia won’t be noticed.

But Africa and South America are small enough so that, as I said, it’s clear, from a globe or satellite photo, what their shape should look like. That was brought out by a cartographer, in the Peters debate, when Peters-advocates tried to weasel out of the matter of shape.

Further, Africa’s and South America’s positions on the equator (but particularly Africa of course, which is more centered on the equator) place them particularly prominently in front of the map-user.

Being on the equator enhances the importance of their shapes for an additional reason: It means that their shape will be well-portrayed in any of the various maps that are conformal at the equator, meaning that Winkel’s or Robinson’s distortion of them, in comparison to those equator-conformal maps’ less-distorted representation, will be more blatantly obvious.

Is Africa more important than South America in those regards? Maybe a little, for two reasons:

1. As I mentioned above, Africa is more centered on the equator, so that what I said above about equatorial-ness applies even more to Africa, as compared to South America.

1a) That matters because, as I mentioned, comparison to a map that’s conformal at the equator makes it more difficult for Winkel or Robinson to get away with distortion there.

1b) Equatorial centering places a continent more prominently in front of you.

2. Many world maps have longitude 0, the Greenwich meridian, as their central meridian. That places Africa more in the middle of the map. That makes a difference for two reasons:

2a) It places Africa more prominently.

2b) If the map has curved meridians, they’re usually less curved near the central meridian, with the result that a continent near that meridian will be more un-distorted. Again, that comparison with that more un-distorted representation (on maps that don’t distort the equatorial regions) makes it more difficult for Winkel and Robinson to get away with their shape-distortions.

So, I don’t mean to show favoritism for Africa. It’s just that, for the above-stated reasons, its distortion on Winkel and Robinson particularly stand out.

Robinson had a lot more considerations in mind than Africa

See above.

…when he created his projection. It just may be that his eye for geography was a lot more trained than yours, and the myriad distortions I keep point out…

One, actually; “Areal infidelity”. You also claim that long distance ruler-measurements on the map are more accurate on Winkel, and I suppose we could call that a distortion-comparison too, bring the total up to two. I’ll comment on your long-distance ruler measurement claim below in this reply. (Briefly, measurement on a map is a silly and inaccurate way to determine great-circle or loxodrome distances).

…that you keep claiming are not noticeable were in fact noticeable to him, to me, to map publishers, to geographers, and to many other people.

Then maybe Winkel is for highly-sophisticated people like you, who better notice a little areal-infidelity, and who understand the great importance of areal-fidelity (I must admit that its great general importance escapes me).

It’s fine you have preferences. It’s not fine that you insist those preferences are shared by “most people”

Wrong. I said that I can’t prove what I said about other people’s preferences. But I supported my claims in numbered statements above in this reply. And I said that one would hope that the experts’ sophisticated knowledge of what matters for map users is informed by surveys of map users.

or that they have some greater objectivity or legitimacy than other preferences. They don’t.

In some numbered statements above in this reply, I supported some claims of mine. I suggest that my numbered arguments were reasonable, and even uncontroversial.

So, yes, I _do_ claim that those arguments have some objective validity.

Additionally, though, sure, I’ve also been making (clearly-labeled) expressions of subjective impressions.

But let me be explicit, here, about my message to other map-users:

If an authoritatively-touted map seems ugly, overly shape-distorted, lacking in justification, or otherwise objectionable…maybe it is.

…despite assurances that it possesses some subtle, esoteric, arcane, but important, merits that you’re just not sophisticated enough to notice or appreciate the great importance of.

When people say that a problem doesn't have an exact solution, they usually mean that it doesn't have a solution that can be written in closed form… When someone tells you that a solution requires the use of numerical methods, they don't mean that you have to press the square root key.

Given that I have a formal education in numerical methods and thirty-five years’ experience in the topic, I’m not inclined to debate this stuff. You meant “closed-form”, not “exact” , it turns out

Yeah here are some links to references about that. Below each link is a quote of a relevant passage not too far from the top of the page linked to:

http://eqworld.ipmnet.ru/en/solutions.htm

“Exact (closed-form) solutions to mathematical equations play an important role in the proper understanding of qualitative features of many phenomena and processes in various areas of natural science.”


http://ocw.usu.edu/Civil_and_Environmen ... Matlab.pdf

“We will present the Newton-Raphson algorithm, and the secant method. In the secant
method we need to provide two initial values of x to get the algorithm started. In the
Newton-Raphson methods only one initial value is required.
[continuing the quote in its next paragraph]

Because the solution is not exact, the algorithms for any of the methods presented herein
will not provide the exact solution to the equation f(x) = 0, instead, we will stop the
algorithm when the equation is satisfied within an allowed tolerance or error, ε. In
mathematical terms this is expressed as
|f(xR)| < ε.”

[conclusion of links and quotes from them]

No doubt you have a different definition of “exact solution”, in sentences like “That equation doesn’t have an exact solution.” I’m not saying that your definition (whatever it is) is wrong. But the way that the above-quoted authors use that term, the way that the authors I’ve read use that term—I suggest that you’re going out on a limb when you say that they’re incorrect.

Usage differs. Many, including expert authors, use “that equation doesn’t have an exact solution” to mean “That equation doesn’t have a solution (‘an answer or, more specifically, a set of numbers that satisfy the equation’) that can be written in closed form, in terms of numbers, symbolically-indicated arithmetic operations (addition, subtraction, multiplication, division, and exponentiation), and symbolically-indicated elementary functions (exponential, trig, hyperbolic, and their inverses).”

Don’t confuse an exact _numerical evaluation_ with an exact _symbolic indication _ of a solution. The square root of two, or the tangent of .53 radians, can’t be exactly written out as a decimal number, but it can be exactly symbolically indicated by 2^.5, or tan(.53)

“Exact” has a dictionary meaning that you can’t really call incorrect. I won’t quote it here, because you can look it up. By that commonly, universally, accepted meaning of “exact”, it’s plain that 1.414, or any further expansion of it, isn’t an exact statement of the square root of two. And it’s likewise plain that 2^.5 is an exact specification of a value that’s the square root of two.

By that universally-accepted meaning of “exact”, a solution in closed form, as described above, is an exact solution. And anything gotten by (e.g.) the Newton-Raphson method, isn’t exact, though it’s closer than was the previous iteration.

Sure, you could say that the instruction to solve an equation by bisection indirectly specifies an exact value of that equation’s solution, because it indicates an (unknowable) value that bisection gets closer and closer to. You could say that such an instruction indicates that exact value as well as does 2^.5, if the equation is x^2 = 2

But it’s understood that, when most people say “an exact solution”, the above paragraph isn’t what they mean.

So let’s not be so critical of definitions other than our own.

Definitions are not solipsistic. Whether or not something affects you does not change the generally agreed-upon definition.

See above, regarding the generally agreed-upon and used definition.

The person who implemented the programming behind that square root key used a numerical method to do it.

See above. No one denies that the square-root key’s programmer used a numerical method. But you yourself don’t do the work of using a numerical method when you press the square root key. He did the work so that you won’t have to.

“Closed-form” is not synonymous with “exact”

Evidently not all authors agree with you.

No one’s criticizing your (unstated) definition of “That equation doesn’t have an exact solution.”

But maybe your definition isn’t the only valid or widely-used one? In fact maybe it isn’t as widely used as the one that you criticize.

I must admit that I have no idea what else “That equation doesn’t have an exact solution.” would mean, other than the meaning used by the above-quoted authors, and stated above by me. But that doesn’t mean that I’m saying your definition (whatever it is) is wrong.

and “numerical method” is not synonymous with “approximation”.

Define as you wish, but a numerical method (when it works) gives an increasingly close approximation to a problem’s solution (defined by Merriam-Webster as its answer or, specifically for an equation, a set of numbers that satisfies the equation).

Whether the numerical method, itself, is the approximation is a philosophical or lexographic issue, on which I tend to agree with you. The numerical method itself is not the approximation. The approximation is what you get when you use the numerical method.

If you want a great-circle distance, or a loxodrome distance, then calculate it rather than measuring it on a map… Instead of stating a nominal scale, intended to be used for the whole map, it would be much better to state the distance, in miles, between the graticule parallels.

Sorry; I find that argument silly.

Sorry, but namecalling, though a common Internet behavior, doesn’t have any value as an argument or way of bolstering an opinion, though it’s often used for that purpose.

North-south distances aren’t more special than other directions.

Actually, on the contrary, north-south distances _are_ “more special” than other directions, on the lat/lon grid.

Here’s the explanation: Parallels aren’t all of the same length. Parallels at higher latitudes are shorter than parallels at lower latitudes. Consequently, the miles per degree of longitude varies considerably, with latitude.

But the meridians are all (nearly) of the same length. What that means is that miles per degree of latitude, measured along a meridian, is nearly the same everywhere on the Earth. (Actually it isn’t entirely invariable, because of the Earth’s oblateness—but close enough for most purposes).

That’s what makes north-south distances, on the lat/lon grid, special. It can be said that a degree of latitude, measured along a meridian, is about 69 miles. …everywhere, with only a little variation at different latitudes.

A map’s “nominal scale” stated at the bottom margin of the map, will be way off in some parts of the map. But, at any graticule quadrangle, if it’s a 10 degree graticule, the distance along a meridian between one graticule parallel and the next, will be about 690 or 691 miles.

And that makes for a better, more useful and accurate scale. Use that distance between two graticule parallels, measured along a meridian, as the map-scale, for measuring or estimating distances in the graticule quadrangle bordered by that measured meridian-distance.


Of course that’s a lot more useful on a conformal projection, because, with a conformal projection, at every point, the scale is the same in every direction.

For measuring world map distances (Route-distances, as opposed to great-circle or loxodrome distances, are the only ones that need to be measured on a map) use a conformal map.

Choosing any other world map for measuring distances (other than a few special distances, from one or a two special points on certain special maps such as Azimuthal Equidistant or 2-Point Equidistant) would be really silly.

I hope that I’ve explained that well.

And this is just another case of having it both ways. If distances are important, you say, then calculate them rather than measure them off a map.

Yes.

But now suddenly some distances—the special ones you like—are important to be able to measure off a map.

Some distances can pretty much only be gotten by measuring on a map. I’m referring to route-distances, along some circuituous route that isn’t a great-circle or a loxodrome.

But if you’re referring to north-south distances between graticule parallels, they _are_ special, and I _do_ like them for their usefulness as scale-indications. They’re certainly useful, and important for that reason, for indicating a relatively accurate scale in the graticule quadrangles that they border.

But that in no way contradicts my suggestion that great-circle distances and loxodrome distances are better calculated than measured.

Maybe you don’t want to calculate a great-circle distance. Or maybe you don’t have a nearby calculator or computer. So measure it on a globe. Maybe you don’t have a globe either. Then use a calculator, computer, or globe at the library, or at school, or at a friend’s house.

Of course, if you’re near a computer, the Internet has distance-calculating websites.

But suppose that you aren’t near a calculator, computer or globe, but you _are_ near a map. You want the distance right away, instead of waiting till you can get to a calculator, computer or globe. So you measure it on the map. For the sake of your measurement’s accuracy, let’s hope your map uses a conformal projection. To measure a distance in a certain graticule quadrangle, judge the scale in that quadrangle by measuring, as I described above, the distance along a meridian bordering that graticule quadrangle, between the graticule parallels bordering that graticule quadrangle.



(Below, the discussion is about Winkel’s long-distance ruler-measurement of distances being more accurate than those made on other maps)

Better than Goode's Homolosine, or Interrupted Sinusoidal?

Interruption is yet another kind of distortion. Those gaps are distances. You can’t just pretend they don’t exist.

…but you can correctly say that they aren’t part of the map. They aren’t distortions, because they aren’t even part of the map. That’s been pointed out by cartographer authors.

Dealing with them implies knowing the great circle route.

Certainly, as you suggest, any route that crosses an interruption will be difficult to follow on the map, won’t be as clear as it would be on an uninterrupted map. But that can be dealt with somewhat. Some interrupted school-maps have numbering along the interruptions, so that a route can be picked up at the other side of the interruption. Of course you must ensure that the resumed route crosses the interrupting meridian at the same angle as it did on the other side of the interruption. But sure, I won’t deny that routes that cross an interruption are difficult.

And, anyway, even on an interrupted map, the scale variation is a lot, and so I don’t suggest measurements on interrupted maps as a good way to determine distances.

If you want a route-distance, or if you want a quickly measured or estimated great-circle or loxodrome distance that uses only a nearby map, then use a conformal projection. Winkel Tripel isn't a conformal projection, and therefore would be a poor choice for measuring distances.

I have no idea what you mean by a “nearby map”

A “nearby map” is a map that isn’t very spatially-distant from you.

In particular, I’m referring to a map that is significantly nearer to you, and more convenient for you to get the use of, than any calculator, computer, or globe.

but this argument is apples-to-oranges. Winkel tripel is near the best for a random selection of distances of random lengths.

But measuring great-circle or loxodrome distances on a world map would really be silly. It’s a ridiculously inaccurate way of determining distances.

Conformal maps are useless for long distances.

Incorrect.

Conformal maps are the only ones worth considering for measuring long or short distances on a world map. (But as I said, the only distances that should be measured on a world map are route-distances, on routes that aren’t great-circles or loxodromes).

Maling pointed out the obvious better suitability of conformal maps for measuring distances on a map (if the mapped region, and the distances measured, are such that the departures from conformality are large enough to make a difference).

That better suitability consists of the fact that, on a conformal projection, at any point, the scale is the same in every direction.

Maling described the obvious, simple, and natural procedure: For any particular region, use the scale in that region. Because the map is conformal, the scale will be the same in every direction at any point (or approximately, in any small region).

A quick easy way:

Estimate the scale in a graticule quadrangle as I described above.

If the route goes into a different graticule quadrangle, then estimate the scale in that graticule quadrangle in the same way.

For more accuracy:

If the scale can be conveniently calculated for any point on the map, then calculate the scale for appropriate points along the route, to numerically integrate the reciprocal of the scale with respect to distance along the route.

In other words:

:Let x represent distance along the route. Let y(x) be the reciprocal of the scale at x.

Numerically integrate y with respect to x, along the route.

For example, for Newton-Cotes methods, you’d choose equal intervals of distance along the route, evaluating y at equal x–intervals. For Gauss integration, you’d choose the x values at which to evaluate y, in the manner that Gauss calls for.

Michael Ossipoff

[daan]

Sorry; I do not find your rationalizations for your preferences convincing. Come back with data if you wish to be taken seriously in making claims about how this or that kind of distortion is more “noticeable”. Map projection promoters have been making such claims for the past 150 years without managing to convince anyone but themselves. However, with respect to this and statements like it:

You said that Winkel has better areal fidelity--and I say, forgive me,but I didn't notice that. I'm sure it's true, but I just didn't notice it. And isn't noticible wrongness what this is all about?

I point out that educating people does not consist of showing them what they expect to see. It does not matter to me if you or even the general public “prefer” that something look a certain way. Map projections such as van der Grinten and Miller were rationalized by their authors as “not too dissimilar to Mercator” because they were afraid their audience had grown so accustomed to the Mercator look that it would not accept radical departures from the familiar. There is no difference in substance in your claims. You advocate that something look a particular way while you discount the importance or relevance of the deleterious effects required to achieve that look.

I have stated several times in talks and essays that the best education in maps comes from frequent exposure to many projections and arrangements of them. Your advocacy for any particular projection and arrangement, along with all the rationalizations, falls on my deaf ears. What you advocate is not educations; it is inculcation of prejudice. When you tell me that you don’t notice this or that problem on the projection(s) you advocate, I merely feel sorry for you.

I don't claim to use or encounter all map projections. But, with the exception of Robinson's, the specifics of all projections can be found, are available to the public.

Not so. Dietrich (Grundzüge der allgemeinen Wirtschaftsgeographie', 1927) published…. Bertin… published…. Many 16th century maps….

Thankfully they aren’t among the maps that we frequently encounter.

This illustrates another reason why I have lost interest in most of this conversation. You were simply wrong, but rather than discard the claim you rationalized on the basis of your faulty assertion, you chose to discount the importance of the contrary evidence. I will make it more clear. You were grossly wrong. Tens of millions of schoolchildren were exposed to Denoyer’s world projection across several decades. That projection was never described publicly until Snyder devised an approximation to it. Note the use of the term “approximation”. It is an approximation because the actual construction method remains unknown.

Which leads to the next. The same trouble crops up in the terminology debate. You trotted out a few sloppy uses of the term “exact” while, of course, ignoring authoritative definitions. There are good reasons why the Wikipedia article on closed-form does not contain the term “exact”, and neither does Wolfram’s site, which is as pithy and authoritative as you can get for mathematical definitions:
http://mathworld.wolfram.com/Closed-FormSolution.html
http://mathworld.wolfram.com/ExactSolution.html

And of course there is the disclaimer about the utility of closed-form expressions:
https://en.wikipedia.org/wiki/Closed_fo ... mputations

And that is where I would like to have gone if the conversation had not been swamped by rationalizations for preferences. You asserted,

There's a significant difference in the kind of work needed for the problems described as needing a numerical method for solution. It means that you can't just punch it out by evaluating a formula….

That is simply wrong, and so wrong that the notion is dangerous in the hands of people who do numerical computing without a proper education in the topic. Closed-form expressions do not imply just “punching it out by evaluating a formula”. Usually you can get away with that over much of the functional domain of the formula, but if you do not concern yourself with the regions of numerical instability and excess cancellation that might be present in the functional domain, then you will produce rubbish results unaware—and some commonly used map projection software packages do just that, and so do many other software packages in other domains. You are no more immune to such problem in closed-form solutions than you are if the solution contain more exotic functions than what are usually “allowed” in closed-form, or than if the solutions use iterative methods.

Meanwhile any commonly used programming language has vast libraries of mathematical functions far beyond those normally thought of as closed-form, functions available to anyone doing numerical work. Hence the distinction of closed-form has little relevance or utility in map projection work or practically any other computational domain. What has relevance is the accuracy of the result and the cost of obtaining it, which is unrelated to whether the solution is “closed-form” or even “exact” by the definition you used.

Sorry, but namecalling, though a common Internet behavior, doesn’t have any value as an argument or way of bolstering an opinion, though it’s often used for that purpose.

There was no name-calling; to call it such is more abuse of definitions. You yourself used the term “silly” to describe practices or arguments no fewer than four times in your latest response. That is three more than I used.

But measuring great-circle or loxodrome distances on a world map would really be silly. It’s a ridiculously inaccurate way of determining distances.

It’s not about measuring distances. That’s not why I brought up distortion of distance. I brought it up because distance is a proxy for all distortions. If the distance metrics are good, then myriad distortion metrics will be good. When I say distance, I do not mean path distance as you kindly but needlessly explicated at length upon. Unqualified distance on globes is great-circle distance.

And just because distance measure will not yield usable accuracy does not thereby absolve the projection from any concern for the matter. If someone looking at Projection A thinks the distance from M to N is three times its actual value, but looking at Projection B thinks it’s twice its actual value, then Projection B has done a better job for that, and if Projection B does a better job than A more than half the time then Projection B has yielded an important service over Projection A. The projections you advocate are not particularly good at that and cannot be due to their straight parallels.

Best,
— daan

[RogerOwens]

Hi daan:

(Yesterday I turned the computer on in order to send this post, and I then noticed your most recent post. At first I intended to combine this post with my reply to your most recent, and began the reply first. Then, halfway through the reply, I decided that I should post this distance information first, and then, in a subsequent separate post, resume and send my reply to your most recent post)

Distances in Winkel and Eckert III:

You said:

Winkel tripel is near the best for a random selection of distances of random lengths.

Let’s check.

But first, let’s take at least a glance at Winkel (by which name I refer to NGS’s vertically-expanded Winkel Tripel):

Look at Winkel’s U.S. Notice how horizontally squashed it looks. No, NGS’s vertical expansion didn’t just uglify Winkel—It also spoiled the good distances that Winkel is so renowned for. We’ve already looked at Africa—same thing. In fact all the low latitudes, and all or nearly all of the middle latitudes are strongly distorted in that way.

How can anyone think that Winkel’s big 1-dimensional expansion would leave good distances, good scale-consistency intact?

The north-south scale and the east-west scale are dramatically, ridiculously, different from eachother. Can anyone call that “good scale-consistency” or good for consistent distance-measurement accuracy?

Now, the measurements:

I compared, on Winkel and Eckert III, nine distances. They included measurements that were horizontal, vertical, diagonal, through the center, not through the center, and peripheral, on the outer meridian.

I’m going to compare, here, the factors by which the average scale along the measured distances differs from the map’s “average scale.”

By a particular map’s “average scale”, I refer to the scale by which that map’s area, expressed in square scale miles, would be numerically equal to the area of the actual Earth, expressed in square miles.


In all but two of those nine measurements, The average scale along the measured distance on Eckert differed from Eckert’s average scale by a smaller factor than the factor by which the average scale along the measured distance on Winkel differed from Winkel’s average scale.

One of those two measurements in which Winkel did better was the distance along the entire central meridian. But, with the distance along the entire equator, Eckert III did better.

The other measurement in which Winkel did better than Eckert III was the distance, along the outer meridian, between lat 60 and lat 90. …a little-inhabited latitude-region. I suppose that measurement could be relevant to a polar expedition.

What I speak of here is reciprocal scale, miles per centimeter. I’ll refer to it as 1/scale.

With Eckert III, of course the scale along the centeral meridian and the scale along the equator are equal. Not so in Winkel, where they differ by a factor of about 1.22

With Eckert III, those two distances’ 1/scale is about 1.18 times Eckert III’s average 1/scale.

With Winkel, the 1/scale along the equator is about 1.31 times Winkel’s average 1/scale. The 1/scale along Winkel’s central meridian is about 1.07 times Winkel’s average scale.

Those two numbers differ by 1.18, in opposite directions, by about the same factor. Of course it goes without saying that a particularly large scale error somewhere is more important than a particularly small scale error somewhere. So, even just in those two measurements, Eckert III does better.

Eckert did better than Winkel along the outer meridian from lat 0 to lat 30, and from lat 30 to lat 60.

Eckert did better than Winkel overall along the entire outer meridian.

The following positions are designated by (lat, lon), where east lon and north lat are positive:

For the following distances:

(60, -90) to (-30, 30)…Eckert better

(60, 30) to (-30, 150)…Winkel only 1 percent better

(60. -90) to (-60, 90)…Eckert better

Let me state these distances more concisely in a list:

1. Along outer meridian:
…lat 60 to lat 90………………..Winkel better
…lat 30 to lat 60………………...Eckert better
…lat 0 to lat 30………………….Eckert better

2. Along all of outer meridian…..Ekert better

3. Along all of equator………….Eckert better

4. Along all of central meridian…Winkel better

5. (60, -90) to (-30, 30)………….Eckert better

6. (60, 30) to (-30, 150)…………About the same

7. (60, -90) to (-60, 90)………….Eckert better

I report here every measurement that I made. I didn’t reject any measurement results. I didn’t “cherry-pick”).

When Eckert III was better than Winkel, Eckert III was usually significantly better.

…And Eckert III is a simple linear projection, not claimed to be optimized for anything. (but still conformal at the equator).

But, is this really surprising? Look at Winkel. Good ruler-distance-measurements need consistent scale, low scale variation. So, look at Winkel—Do you notice anything about it that is inconsistent with consistent scale? How about the ridiculous difference between north-south and east-west scales on most of the map.

Then how is it that cartographers think that Winkel is “near the best” for ruler-measured distances?

How about this explanation?:

The praise that’s heaped on Winkel was originally given to Winkel Tripel _without the vertical expansion_. In the manner of a rumor, that praise carried over to Winkel after the expansion/uglification had spoiled its good distances.

Winkel Tripel is usually described as an averaging of Cylindrical Equidistant (CE) and Aitoff. Usually that’s all that’s said. But of course, in the form that NGS uses Winkel, the form that Winkel Tripel is nearly always shown in, there is large vertical expansion of the map.

I don’t know who perpetrated that expansion—whether it was Winkel himself, or NGS, or someone inbetween.

Of course an averaging of CE and Aitoff will have an aspect-ratio of 2:1 It, too, will be conformal at the equator. And it could be expected to have particularly low scale-variation (resulting in good ruler-distance-measurements, as if anyone wanted to measure great-circle distances in that way).

If I refer to it again, I’ll refer to Unexpanded Winkel-Tripel as “WTU”.

I have no criticism of WTU. It isn’t ugly; its appearance isn’t ridiculous.

It likely has relatively low scale-variation, but it gives up the pseudocylindrical convenience of location-determining, lat/lon X/Y conversion. That’s a trade-off, and your choice would depend on what you want. WTU might also have better shapes, overall, than Eckert III.

I don’t advocate either of Eckert III or WTU over the other, for use in atlases or as wall-maps.

Eckert III, WTU, Aitoff, Hammer-Aitoff, or Quartic would be fine for an atlas or a wall-map.

If one wants equal area and pseudocylindrical-ness, then Eckert IV or Mollweide would be ok. Yes, they have some funny shapes, but at least it’s for a reason, and something (equal-area and pseudocylindricalness) is gained thereby.

Quartic achieves that gain (equal-area and pseudocylindricalness), without the funny shapes, but Quartic has unusually wide aspect-ratio, and particularly low space-efficiency.

As I mentioned, those disadvantages can be mitigated by otherwise filling and making good use of the corner-space that the map doesn’t use, on the book-page or the map-sheet.

Michael Ossipoff

[daan]

Michael,

See, for example, Goldberg & Gott p. 31, where they give metrics (including distance) for both forms of Winkel tripel. The form recommended by Winkel (which is what NGS uses) performs better for distance measurements than the form used by Times. Six of one, half dozen of the other, for the other metrics.

Best,
— daan

[RogerOwens]

See, for example, Goldberg & Gott p. 31, where they give metrics (including distance) for both forms of Winkel tripel. The form recommended by Winkel (which is what NGS uses) performs better for distance measurements than the form used by Times. Six of one, half dozen of the other, for the other metrics.

Yeah I think RMS is the culprit.

G&G use a standard error-aggregation formula. Ordinarily, it’s a good one. It uses RMS. RMS has all sorts of justifications, and favors the reduction of larger errors, even at the cost of increasing smaller errors by a greater amount. Ordinarily that would be a good thing--except when that favoured large-error-reduction occurs in regions of a world map that are relatively uninhabited, little-noticed, and not prominently-positioned…at the expense of more important and noticeable regions.

Thus, we get unnecessary heavy drastic shape-distortion in low and middle latitudes, even at the map’s center, and more distance-inaccuracy in non-polar regions, as compared to Eckert III

…and a high G&G rating.

In regards to disparity between E-W and N-S scales, even at the center of the map, NGS Winkel is 2nd only to Peters.

At least Peters is conformal halfway between equator and pole (NGS Winkel remains drastically E-W compressed in middle latitudes—look at its U.S.). And at least Peters gets equal-area for its shape distortions.

I admit that my distance-measurements were mostly over non-polar land, explaining why Eckert III did better than NGS Winkel.

I don’t criticize people for valuing different things than I value. Some people like NGS Winkel because some scientists say that it did well by their formulas. Others might not prefer it because of its unusually drastic shape-distortion over most of the Earth, and its losing to Eckert III in a broad representative set of (admittedly mostly non-polar) distance-measurements (unsurprisingly, given its unusually high N-S/E-W scale-disparity).

As I said before, “To each their own”.

I wasn’t comparing NGS Winkel to Times Atlas Winkel. I was comparing NGS Winkel to Eckert III.

As for the Times Atlas’s Winkel Tripel, I don’t know what its vertical (Y dimension) expansion-factor was (relative to the un-expanded 2:1 aspect-ratio Winkel-Tripel, the averaging of Square-Grid CE and Aitoff). But I presume that its expansion-factor was less than that of NGS Winkel.

A website said that the Times Atlas’s Winkel Tripel had a “standard parallel” of lat 40. …whatever that means. I looked up “standard parallel” at several websites, but only found these definitions: The parallel where the map contacts the globe. The parallel at which there’s no distortion . The parallel along which the scale equals the nominal scale.

If they’re saying that the Times Atlas’s Winkel Tripel is conformal where the 40th parallel meets the central meridian, then that wouldn’t be so bad. At least N-S scale and E-W scale would be similar in mid-latitudes. …which is more than can be said for NGS’s Winkel.

Maybe the latitude of Winkel’s residence was somewhere in the 50s?

But, Times Atlas Winkel isn’t WTU, and so Goldberg & Gott haven’t compared NGS Winkel to WTU either, just as they haven’t compared it to Eckert III.

The “WTU” that I referred to is the actual averaging of square-grid CE (aspect-ratio 2:1) and Aitoff, resulting in a map with an aspect-ratio of 2:1. I expect that it _does_ have good shapes and distances I don’t deny that it likely has better shapes, overall, than Eckert III. Of course it’s conformal at the equator, as is Eckert III.

Though it’s inadvisable to determine great-circle distances from map-measurements, they still have some relevance, because some maps give a more accurate impression of distances than others do.(even though the map-distances are rarely accurate).

Besides, it’s a measure of average scales along those measured lines, and scale is relevant to one’s impression of short distances on the map too.

Sorry; I do not find your rationalizations…

Definition: A rationalization is a justification or reason for a statement or position with which the speaker disagrees
.

…for your preferences convincing. Come back with data if you wish to be taken seriously in making claims about how this or that kind of distortion is more “noticeable”.

How about if cartographers come back (and only then make recommendations to atlas-publishers) when they have polling and survey data regarding what people prefer.

I don’t claim to have that information. But here’s a hint: Why do so few stores stock Robinson &/or NGS Winkel?

[/quote]
Sure, survey/polling data would be helpful, but surely you must agree with some of the things that I said, in my four numbered arguments regarding shape vs area.

For example:

1. Isn’t it true that a small shape distortion is a lot more noticeable, to you as well, than a small area-error?

2. When something is described as “ugly”, whether a sculpture, a car, or forms in a painting, how often is “ugly” used to refer to its size rather than its form or shape?

3. How important is area-relation-accuracy between two shape-distorted blobs, best identifiable by their position and labeling? (Yes, I know, NGS Winkel’s shapes aren’t _entirely_ unrecognizable, though maybe close to it (e.g. Look at its U.S.). But, as I said, it’s 2nd only to Peters in that regard).

4. Do you think that, in Eckert III, the magnification of Greenland is achieved at the expense of Ecuador’s fair size or shape?

But never mind this shapes-vs-areas issue—We’re talking about an extreme. NGS Winkel’s
N-S/E-W scale disparity is unnecessary, unusual, and ridiculous.

People might draw different conclusion. Those four numbered statements (worded here as questions) might not lead you to the conclusions that they point to for me. But those four statements I made, regarding shape vs area, are probably not really so easy to justify disagreement with.

You said that Winkel has better areal fidelity--and I say, forgive me, but I didn't notice that. I'm sure it's true, but I just didn't notice it. And isn't noticeable wrongness what this is all about?

I point out that educating people does not consist of showing them what they expect to see. It does not matter to me if you or even the general public “prefer” that something look a certain way.

…like what it’s supposed to represent? :^)

I thought we were talking about aesthetics, not education.

So cartographers want to educate people’s preferences to match the current preferences/fashions of cartographers?

Aesthetics is about what people prefer, and that’s an individual subjective matter.

Because there are sure to be wrong measure-quantities on any world map (distances, and shapes & directions or areas), then no flat map accurately educates, because some of it is sure to be wrong.

Because it’s just a choice of _what_ should be right, that’s an aesthetic matter. …unless there’s practical need for a certain property, such as cylindroid-ness (pseudocylindricalness or cylindricalness) in a practical data map, or equal-area in a map intended for graphically showing what percentages of the Earth have the various kinds of vegetation, or conformality for navigation (or just for people interested in distances and directions, or people whose aesthetic preference is good shapes).

It’s regrettable that Robinson and NGS Winkel are the only noncylindrical world maps available for sale, and are the only maps featured in atlases, as the atlas’s main world map.

I don’t think that Robinson’s and Winkel’s low sales means that people wouldn’t like other non-cylindrical maps, such as Hammer-Aitoff, Aitoff, Eckert III, or Un-Expanded Winkel Tripel (Which I call “WTU”). …none of which have NGS Winkel’s ugly, unnecessary shape-distortion.

You advocate that something look a particular way [like what it’s supposed to represent?] while you discount the importance or relevance of the deleterious effects required to achieve that look.

We all discount the relative relevance, importance, of, or need for, what we’d give up in order to gain something else. I’ve told reasons to support my suggestions on that. I gave four arguments regarding area vs shape, and then put those considerations into the form of four questions. With Winkel, we’re talking about an extreme. I discussed NGS Winkel’s unusual, unnecessary extreme, as regards N-S/E-W scale-disparity.

I have stated several times in talks and essays that the best education in maps comes from frequent exposure to many projections and arrangements of them.

Of course. No objection to that. That’s another reason why it’s regrettable that only Robinson and Winkel can be found in atlases, as the atlas’s primary world map, and that they’re the only noncylindrical maps available for sale as wall-maps I agree that it would be nice to have more variety and selection in that regard.
.
(As I mentioned at the top of my previous reply, when I say “Winkel”, I’m referring to NGS’s vertically-expanded Winkel-Tripel. As for Unexpanded Winkel Tripel, I call it WTU. I have no criticism of WTU. It isn’t ugly, or unnecessarily, drastically shape-inaccurate, as Winkel is, and its non-polar distances are surely better than those of NGS Winkel.)

Your advocacy for any particular projection and arrangement, along with all the rationalizations, falls on my deaf ears.

I suggest that deaf ears aren’t the best way to find out what customers prefer. …But then, you said that you don’t care what the public prefer. Do you clarify that when recommending to atlas-publishers?

And, for the atlas-publishers themselves, unless most of their customers are cartographers, maybe cartographers’ fashions aren’t the best thing for publishers to go by.

What you advocate is not educations; it is inculcation of prejudice.

I fully admit that what I advocate isn’t education.

As for inculcating prejudice, I had to look up “inculcate”, and Meriam-Webster said it means to teach, especially by repetition or admonition. I’m not advocating teaching people what to like. People already have preferences, and I’m not advocating trying to teach them different preferences. What I advocate is that map-customers and potential map-customers be listened-to, and that their preferences be respected by atlas-publishers.

If you call their preferences “prejudices”, that sounds elitist. So is the notion that it’s necessary to educate people so that their preferences match those of cartographers, among whom Robinson and Winkel are currently the big fashion.

When you tell me that you don’t notice this or that problem on the projection(s) you advocate, I merely feel sorry for you.

(“This or that problem” refers to the slight difference in high-latitude magnification, between Eckert III as opposed to Robinson and Winkel.)

…Then you must feel very sorry for yourself, for not noticing Winkel’s unusual, ridiculous, humungous disparity between E-W and N-S scale throughout low and middle latitudes.

Do you think we could refrain from sharing our opinion of those with whom we disagree?

“Internet” needn’t by synonymous with “Flamewar behavior”.


Here is what I said I didn’t notice: I said that I don’t notice that Winkel and Robinbson don’t magnify Greenland as much as Eckert III does.

Looking at an equal-area map, the part of South America north of the equator (I’ll call it “North South-America”) looks just slightly bigger than Greenland.

What is noticeable at a glance is that Eckert III, Robinson, and Winkel all seem to show Greenland very roughly twice the size of North South-America. That’s just at a brief first glance.

In other words, any difference between the Greenland-magnifications of Eckert III, Robinson, and Winkel doesn’t stand out, at first brief glance, as dramatically different. Most would agree, then, that those differences aren’t aesthetically odious.

Now, on close examination, when a difference is looked-for, Eckert III might magnify Greenland just a little more than Robinson and Winkel do.

One could actually determine what percentage of the map’s world area is given to Greenland (or the 30-degree graticule-quadrangle that covers most of Greenland), by Eckert III, Robinson, and NGS Winkel.

Maybe, by lat/lon to X/Y formula, or by map-measurement, measure that area by transects.

But that would hardly be necessary—Aesthetics isn’t about numbers.

Still, anyone wanting to use Eckert III’s, Robinson’s and NGS Winkel’s high-latitude magnifications as an argument for Robinson and Winkel versus Eckert III, should cite those numbers.

Thankfully they [some undefined projections you named] aren’t among the maps that we frequently encounter.

You were simply wrong

No, I didn’t say that never in history has an undefined projection been widely-encountered. My statement explicitly referred to the _current_ situation—maps currently frequently-encountered.

Yes, I admit that the parameters needed for conversion between lat/lon and X/Y regrettably aren’t given with Azimuthal Equal Area maps in atlases. (even when used as data-maps). But at least the projection itself is known, and, if someone really wanted to, those parameters could be determined by solving a system of nonlinear equations.

Which leads to the next. The same trouble crops up in the terminology debate. You trotted out a few sloppy uses of the term “exact”

Definition of a “Sloppy Usage”: A usage different from the one that the speaker believes is the uniquely correct one.

You know that this is a peripheral issue, without relevance to the matter of Robinson & NGS Winkel vs Eckert III, Times Winkel, Unexpanded Winkel (2:1 aspect-ratio), Aitoff, Hammer-Aitoff, Quartic, etc.

…while, of course, ignoring authoritative definitions.

I don’t ignore your definition, or say that your definition is wrong. I’m not the one making an issue of those words’ definitions.

I merely said that those quoted authors’ usage is widespread. As I said before, let’s not be so critical of usages other than our own.

There's a significant difference in the kind of work needed for the problems described as needing a numerical method for solution. It means that you can't just punch it out by evaluating a formula….

That is simply wrong, and so wrong that the notion is dangerous in the hands of people who do numerical computing without a proper education in the topic. Closed-form expressions do not imply just “punching it out by evaluating a formula”. Usually you can get away with that over much of the functional domain of the formula, but if you do not concern yourself with the regions of numerical instability and excess cancellation that might be present in the functional domain, then you will produce rubbish results unaware

Ok, but I don’t usually (ever?) encounter closed formulas so elaborate that they’re computationally problematic as you describe. (…but maybe I would have, if I’d done more calcuation involving map projections.). I merely meant that it’s easier and quicker, for example, to calculate distance along a circle, than to calculate distance along an ellipse. …Easier to solve a small equation whose solution is writeable in closed-form, as opposed to an equation requiring Newton-Raphson, Bisection, etc.

But measuring great-circle or loxodrome distances on a world map would really be silly. It’s a ridiculously inaccurate way of determining distances.

It’s not about measuring distances. That’s not why I brought up distortion of distance. I brought it up because distance is a proxy for all distortions. If the distance metrics are good, then myriad distortion metrics will be good.

You mean like the way NGS Winkel’s big disparity in N-S distance-scale vs E-W distance-scale results in real funny shapes?

And just because distance measure will not yield usable accuracy does not thereby absolve the projection from any concern for the matter. If someone looking at Projection A thinks the distance from M to N is three times its actual value, but looking at Projection B thinks it’s twice its actual value, then Projection B has done a better job for that, and if Projection B does a better job than A more than half the time then Projection B has yielded an important service over Projection A.

Agreed, as I mentioned earelier in this reply. It can be a good thing for a map to give better impressions of distance, compared to another map.

The projections you advocate are not particularly good at that and cannot be due to their straight parallels.

As I said, Eckert III did better than NGS Winkel in that regard, over a broad representative set of distances (admittedly mostly not over polar land).

Evidently, for the purpose of good distances, NGS Winkel’s huge disparity between N-S scale and E-W scale did more harm than Eckert III’s straight parallels.


Not all of the projections that I advocate have straight parallels. I also advocate Times Atlas Winkel Tripel, Unexpanded Winkel Tripel (aspect ratio 2:1), Aitoff, Hammer-Aitoff, Lagrange, August, your Oblated Lagrange, etc.

Michael Ossipoff

[RogerOwens]

A few more comments:


I was pointing out the obvious fact that NGS Winkel drastically east-west compresses most of the Earth. I should be a little more specific. NGS Winkel’s U.S. is severely distorted in that way, and the U.S. extends up to lat 49. (The U.S would be distorted in that way even if it were at the map’s central meridian).

Between lat 49 and lat -49 is about ¾ of the Earth’s surface area. So NGS Winkel drastically east-west compresses about ¾ of the Earth’s surface.

But, effectively, of course it’s more than that, isn’t it:

South of lat -49, there’s very little land other than Antarctica. So, if the amount of inhabited land south of lat -49 is added to the area north of lat 49, that only adds up to about 1/8 of the Earth’s surface—the region that isn’t so E-W compressed by NGS Winkel (and where NGS Winkel might even bring improvement).

So, among that 7/8 of the Earth’s surface that we have when we leave out most of the area south of lat -49, 6/7 of it is drastically E-W compressed.

Only 1/7 of that 7/8 isn’t as distorted as the U.S. .

How could we expect the expansion of Unexpanded Winkel Tripel, into NGS Winkel Tripel, to affect distances and scales?:

We can’t speak of Max/Min scale, because both Winkels have infinite E-W scale at the polls. But we can speak of Max/Min _finite_ scale, which ignores that one infinite scale at that one point. Will the NGS expanded Winkel have more, or less, Max/Min finite scale, as compared to Unexpanded Winkel? It will have more:

A horizontal measurement at Winkel Tripel’s center won’t be affected by the N-S expansion. Any measurement along the outer meridian will be lengthened by the N-S expansion.

So, if the max finite scale is somewhere on the outer meridian, and if the min scale is at the map’s center, then NGS Winkel’s N-S expansion must increase the Max/Min finite scale.

Anyway, it’s obvious that the 1-dimensional expansion is making scales more different from eachother (N-S and E-W scales).

I’m pleased to report that the Oxford Atlas of the World uses Unexpanded Winkel Tripel.

When it it better to favor good shapes at the equator (as does Unexpanded Winkel), and when is it better to favor good shapes at middle latitudes or a compromise among shapes at various latitudes (as does Times Atlas Winkel)?

I suggest that mid-latitude shape, or a mid-latitude compromise among the world’s latitudes, is more important if there will be a need to use that world map for examination of regions, and it isn’t known which regions will have to be examined. I suggest that that situation exists when the world map is the only map conveniently available nearby. So the world map’s portrayal of regions is important, as opposed to only its showing of the relative positions of regions, continents, etc.

I suggest that that situation typically exists for a wall map, or for any book (say geography or history) that has a world map, but doesn’t have the space, or choose to use the space, for regional maps.

I suggest that, in an atlas, where regional maps are available, a world map’s portrayal of regions isn’t so important, in comparison to its showing of the positional relation of regions…and in comparison to its aesthetic appearance. I’ve been claiming that equatorial shapes, and, in particular, the shapes of the large continents intersected by the equator, are particularly important for aesthetic appearance of a world map, at least partly due to their more prominent positioning.

Among Winkel-Tripel versions, that suggests that Unexpanded Winkel would be better for atlases, and that Times Atlas Winkel would be better for most other applications.

Michael Ossipoff

[RogerOwens]

Oops--With a flat-pole projection, there's no such thing as maximum finite scale, because you can find an east-west scale as large as you want, by getting sufficiently close to the pole.

I should have said "not arbitrarily increasing" scale.

All 3 versions of Winkel Tripel have that arbitrarily increasing E-W scale as the pole is approached--so they're similar in that regard. They differ in their maximum scale elsewhere, and that was what I meant. Other than the arbitrarily-increasing E-W scale as the pole is approacxhed, NGS Winkel Tripel has a larger Max/Min scale than do Times Atlas Winkel Tripel and Unexpanded Winkel Tripel. That was what I meant.

Michael Ossipoff

[daan]

Definition: A rationalization is a justification or reason for a statement or position with which the speaker disagrees

Use “justification” in place of “rationalization” if it makes you feel better. “Rationalization” isn’t inherently derogatory and is even positive in some senses. I don’t find the justifications you give for your preferences convincing.

rationalize |ˈra sh ənlˌīz; ˈra sh nəˌlīz|
verb [ trans. ]
1 attempt to explain or justify (one's own or another's behavior or attitude) with logical, plausible reasons, even if these are not true or appropriate : she couldn't rationalize her urge to return to the cottage.

In regards to disparity between E-W and N-S scales, even at the center of the map, NGS Winkel is 2nd only to Peters.

I can’t comment on such claims when you haven’t even defined the universe of projections for which you intend your statement to apply. It certainly isn’t true across all projections.

A website said that the Times Atlas’s Winkel Tripel had a “standard parallel” of lat 40. …whatever that means. I looked up “standard parallel” at several websites, but only found these definitions: The parallel where the map contacts the globe. The parallel at which there’s no distortion . The parallel along which the scale equals the nominal scale.

It’s a questionable usage of “standard parallel”. What it means with regard to all forms of the Winkel tripel is the parallel at which the map is conformal at the central meridian.

Maybe the latitude of Winkel’s residence was somewhere in the 50s?

Winkel chose the standard parallel that yields the correct overall area for the map. It’s ±50°28′, hundreds of miles north of his residence.

Of course it’s conformal at the equator, as is Eckert III.

It’s also grossly disproportionate in area, as is Eckert III.

How about if cartographers come back (and only then make recommendations to atlas-publishers) when they have polling and survey data regarding what people prefer.

I addressed this. There was a time when people preferred Mercator maps. Education does not consist of pandering to prejudice.

1. Isn’t it true that a small shape distortion is a lot more noticeable, to you as well, than a small area-error?

No—but a qualified no, since there is no uniquely correct way to compare magnitude of shape distortion to magnitude of disproportion in area. Given this discussion, it’s likely I’m more sensitive to area disproportion than you.

2. When something is described as “ugly”, whether a sculpture, a car, or forms in a painting, how often is “ugly” used to refer to its size rather than its form or shape?

This analogy is inapt. It’s not about absolute size. It’s about disproportion, and certainly people notice disproportion and even think it grotesque in some contexts.

4. Do you think that, in Eckert III, the magnification of Greenland is achieved at the expense of Ecuador’s fair size or shape?

No. The magnification of Greenland happens because the meridians don’t converge quickly enough toward the poles.

But those four statements I made, regarding shape vs area, are probably not really so easy to justify disagreement with.

I already have, repeatedly. You are willing to salve your preferences by making sacrifices that other people find more objectionable than the problems you ameliorate. That’s fine. I repeat, you need real data for your claims that your preferences are shared by most or even many people, and even with such data, given that those preferences are mere prejudices, their relevance to a responsible choice of map projection is questionable.

I thought we were talking about aesthetics, not education.

I didn’t. I thought we were talking about how the Robinson projection’s interpolation method was defined, and in the absence of any standard method, whether the Robinson projection could even be called a map. You seem to have abandoned that particular conversation. Then there was some digression about why NGS chose Winkel tripel, at which point you began making many claims about æsthetics, the value of which claims I disputed. I’m not interested in debating what’s prettier; while I have my own preferences, I would no more debate those than I’d debate food preferences.

So cartographers want to educate people’s preferences to match the current preferences/fashions of cartographers?

Please stop raising straw men. Being acquainted with an awful lot of cartographers, I assure you there is no solidarity in preference or recommendation amongst them. The dynamics of map projection selection in world reference maps go far beyond the poor sot who ends up with the contract.

Aesthetics is about what people prefer, and that’s an individual subjective matter.

Then stop trying to make them a matter of debate.

But here’s a hint: Why do so few stores stock Robinson &/or NGS Winkel? … It’s regrettable that Robinson and NGS Winkel are the only noncylindrical world maps available for sale, and are the only maps featured in atlases, as the atlas’s main world map.

I cannot reconcile those two statements.

I have stated several times in talks and essays that the best education in maps comes from frequent exposure to many projections and arrangements of them.

Of course. No objection to that. That’s another reason why it’s regrettable that only Robinson and Winkel can be found in atlases, as the atlas’s primary world map, and that they’re the only noncylindrical maps available for sale as wall-maps I agree that it would be nice to have more variety and selection in that regard.

I do not refer to the very limited palette of projection that have been discussed here, and most particularly not to just equatorial aspects.

I suggest that deaf ears aren’t the best way to find out what customers prefer.

I don’t care what customers prefer. My customers can make maps on whatever projection they want.

But then, you said that you don’t care what the public prefer. Do you clarify that when recommending to atlas-publishers?

I have made few recommendations, and when I have, those who asked already had strong constraints on what they would accept for an answer. I don’t tell them their business. I just tell them the possibilities and ramifications.

If you call their preferences “prejudices”, that sounds elitist. So is the notion that it’s necessary to educate people so that their preferences match those of cartographers, among whom Robinson and Winkel are currently the big fashion.

Do you willfully twist what I write? If everyone used Winkel tripel, then that would be inculcating a prejudice. If everyone used Robinson, then that would be inculcating a prejudice. Or Eckert III. Or Fuller dymaxion. Or Mercator. Or Briesemeister. Or only equatorial aspects. Or any particular projection or presentation. World maps are all grossly wrong. To the extent that the public imagines a particular projection looks right, the public is wrong and ignorant.

Here is what I said I didn’t notice: I said that I don’t notice that Winkel and Robinbson don’t magnify Greenland as much as Eckert III does.

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.

What is noticeable at a glance is that Eckert III, Robinson, and Winkel all seem to show Greenland very roughly twice the size of North South-America. That’s just at a brief first glance.

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.

No, I didn’t say that never in history has an undefined projection been widely-encountered. My statement explicitly referred to the _current_ situation—maps currently frequently-encountered.

This is what you wrote. It came with no qualification, and the context suggested no such qualification, let alone anything “explicit” about currency or frequency of encounter:

I don't claim to use or encounter all map projections. But, with the exception of Robinson's, the specifics of all projections can be found, are available to the public.

More importantly, the reason you wrote it was to defend your claim that a projection whose details you do not know cannot even be called a map. However, when pressed, you then stated,

Then what is it about? It’s a suggestion that it would be good to state a map’s projection.

Of course it would be good, but that’s not at all what were discussing. What were discussing was this claim:

means that Robinson's, when found in an atlas or as a published wall-map, can't accurately be called a map.

So please stop changing your claims. They’re right there, recorded for posterity, and I would have hoped by now you would recognize I’m not likely to overlook shifting theses or fabricated context.

Ok, but I don’t usually (ever?) encounter closed formulas so elaborate that they’re computationally problematic as you describe.

Winkel tripel itself is an example, yet it’s pretty simple as projections go. The center point of the projection has to be handled as a special case, and to really do nearby points properly, you have to expand around that as a series. Bonne is a veritable mess for some parameterizations.

Easier to solve a small equation whose solution is writeable in closed-form, as opposed to an equation requiring Newton-Raphson

Even that does not follow. Would you really rather solve a quartic equation in closed form rather than via Newton-Raphson? Or even a cubic, for that matter? Unless you’re careful, a mere quadratic can end up a mess. There are endless examples of problems more easily solved numerically than in closed-form.

— daan

[RogerOwens]

When I speak of “the Arctic”, I refer to the region north of the Arctic Circle. When I speak of arctic magnification, I refer to magnification of the percentage of the Earth’s surface that’s in the Arctic (on the map, vs on the Earth).

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

Some conclusions from that:

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.

2. Eckert III doesn’t magnify the Arctic much more than NGS Winkel does.

But, if the Arctic is regarded as a particularly important area, then it should be mentioned that, if Eckert III, NGS Winkel, and Robinson are scaled so that their circumscribing rectangles have the same area, then Eckert III’s average resolution in the Arctic is 21.6% better than that of NGS Winkel, and 23.1% better than that of Robinson.

So Eckert III’s greater arctic magnification isn’t necessarily undesirable, because, as you said, the Arctic is an important region.

Eckert III’s space-efficiency is .893

NGS Winkel’s space-efficiency is about .82

Robinson’s space-efficiency is about .869

By the way, on NGS Winksl and Robinson, I determined areas by measurement on the maps, by transects and Simpson’s rule. To save time, so as not to make this reply avoidably late, I made as few measurements as possible. So I only used one application of Simpson’s rule for each curved-boundary region that I measured the area of. (But, in NGS Winkel and Robinson, when measuring the area of the Arctic, of course I used separate applications of Simpson’s rule for the regions to the left and to the right of the end of the pole-line, because of the map-boundary’s abrupt direction-change there).

So, on NGS Winkel, and on Robinson, I used only one application of Simpson’s rule, one parabola, to measure the area of a quadrant of the map, from equator to pole. …to minimize the number of measurements needed.

As a result, both on NGS Winkel, and on Robinson, the Simpson’s rule parabola had a maximum in that region of area-measurement (a quadrant of the map). I realize that that should be avoided, probably by using smaller intervals, and more measurements, but, as I said, I wanted to save time.

I emphasize that the Simpson’s rule parabola did _not_ have a maximum in the interval of area-measurement when measuring the area of the Arctic.

Because of the Simpson’s rule parabola having a maximum in that region of area-measurement when measuring the area of a quadrant of the NGS Winkel and Robinson maps, then I’ve overestimated the area of those maps—maybe overestimated it by a lot.

That means that, for NGS Winkel and Robinson, I’ve underestimated the arctic magnification. And it means that, for NGS Winkel and Robinson, I’ve overestimated the space-efficiency.

In other words, the error resulting from the Simpson’s rule parabola having a maximum in the area-measurement interval, when measuring a quadrant of NGS Winkel, and of Robinson, resulted in an unrealistically favorable portrayal of NGS Winkel and Robinson.

I felt that it would be quicker to say all that than to make the 8 additional measurements needed to halve the measurement-interval-size, in an effort to avoid the Simpson’s rule parabola having a maximum in the measurement interval when measuring the area of a quadrant of NGS Winkel and Robinson.

I’ve probably overemphasized the importance of my criticism of Robinson and NGS Winkel. I don’t mean it as social protest, though that’s what I’ve been making it sound like. As you said, nearly all map projections have some advantage, purpose or value. Thanks for pointing out the G&G formula justification for NGS Winkel. I don’t claim that it’s without meaning. I merely question whether it chooses something that we really want, when we look at things other than the result of a formula. When I criticize some map projections, it isn’t, for me, an angry argument. And it’s only the projections that I criticize, not people.

Even the less desirable projections are interesting and fun, if only because of their undesirability, and for discussion of reasons for the curious fact of their popularity with publishers.

So, as meant by me, everything I say here is positive, even when it’s critical of a map projection.

And NGS Winkel isn’t as prevalent as I’ve implied. I don’t really notice it in many atlases. Maybe nearly only in NGS publications. Robinson is the one that nearly all atlases seem to have adopted. But I certainly have no personal complaint about that—I now have a copy of the Oxford World Atlas, whose main world map is Unexpanded Winkel Tripel..

Besides, as you pointed out, nowadays anyone can make a map on any projection that they choose, so it doesn’t really matter that most atlases make a poor choice of a world map.

As you mentioned, I shouldn’t make a contentious issue of aesthetic preferences. 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? :^)

I’m not so sure that there are wrong aesthetic preferences. …or wrong preferences among the various different purposes, uses, advantages and values of different map projections.

(Of course, then, for that reason, I shouldn’t be so critical of the personal preferences of most atlas publishers.)

Because the world is round and maps are flat, there’s no reason to expect a world map to show all regions at the same magnification (…though of course we can make one that does, when proportionately-correct area is important and needed _for some reason_.)

What if a map magnifies some regions? (Eckert III doesn’t do so at the expense of other regions) So what? It means that the map has better resolution in those magnified regions, and that can only be a good thing. More room for legible writing on the map, and for other features to be clearly shown, and more readily examined. What’s wrong with that?

(And, if you compare Eckert III’s better resolution in the artic to NGS Winkel’s alleged arctic advantages, I’ll point out that, in NGS Winkel, those advantages _do_ come at the expense of low and middle latitude regions, around ¾ of the Earth’s surface—it comes at the cost of gross deformation, in the form of big N-S/E-W scale-disparity over ¾ of the Earth’s surface. More about that, farther below where you speak of the Arctic.)

“But Grandmother, what big magnification you have!”

“All the better to examine you with, my dear.”

I don’t understand the objection to magnification.

1In regards to disparity between E-W and N-S scales, even at the center of the map, NGS Winkel is 2nd only to Peters.

I can’t comment on such claims when you haven’t even defined the universe of projections for which you intend your statement to apply.

Yes, I left that out, to keep the sentence brief. I knew that it was vague for that reason, but I felt that brevity was important. But the statement seems true in at least one of the following four ways:

1. Replace “Peters” with: “Peters and other vertically-expanded Cylindrical Equal Area (CEA) projections, and maybe a few other obscure, little-discussed, projections that aren’t often encountered or found even at Internet map-projection websites.

2. The comparison-universe consists of maps that are frequently encountered, currently, in the U.S. (I can’t speak for anywhere else), in Atlases and wall-maps.

3. Same as #2, but add “and textbooks.”

4, Same as #3, but add “and other books, and magazines, newspapers, etc.”

It certainly isn’t true across all projections.

Of course not. Even a brief look at the Internet will show a variety of close-Peters-relatives—other vertically-expanded CEA maps. …some of which, like Peters, are worse than NGS Winkel, in the way I spoke of. …some of which are even worse than Peters in that regard.

What it means with regard to all forms of the Winkel tripel is the parallel at which the map is conformal at the central meridian.

Yes, that’s what I expected that it meant.

That’s why Unexpanded Winkel (Oxford Atlas Winkel) and Times Atlas Winkel are well-justified and well-suited for the applications that I spoke of in my previous posting. ….and NGS Winkel isn’t.

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

With respect to or in reference to what? How did he define “correct area” for that purpose? Any map has the correct overall area, as measured by its average scale, as I defined average scale in a previous posting.

It’s ±50°28′

It’s nice that (at its positive value) it’s in an inhabited latitude.
(I know, you pointed out that the uninhabited arctic is important too, and I’ll answer that below, where you say it.)

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.

How about if cartographers come back (and only then make recommendations to atlas-publishers) when they have polling and survey data regarding what people prefer.

I addressed this. There was a time when people preferred Mercator maps. Education does not consist of pandering to prejudice.

That comment is answered above, in this posting. I don’t think disagreeing preferences can be called prejudice, because that implies that there is objective information by which preferences should be guided. But preferences aren’t like that. They’re subjective. And, for that reason, I don’t agree that people need to be educated to different preferences.

In answer to something that you say farther below, no I’m not intentionally misinterpreting your meaning, and trying to change it. I’m honestly trying to answer what you seem to be saying.

If I mis-interpret and mis-state what you mean, then it’s unintentional, and I apologize in advance.

You can say that a preference is suboptimal in regards to the merit of its results, in some hopefully-objective regard, by some hopefully-objective measure; or in regards to the objective correctness of arguments or justifications, claimed to be objectively-true, for that preference. But, other than that, a preference, as such, doesn’t, in and of itself, have rightness or wrongness, goodness or badness.

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?

1. Isn’t it true that a small shape distortion is a lot more noticeable, to you as well, than a small area-error?

No—but a qualified no, since there is no uniquely correct way to compare magnitude of shape distortion to magnitude of disproportion in area.
Given this discussion, it’s likely I’m more sensitive to area disproportion than you.

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.

Alternatively, of course there is an obvious way to compare distortions of area, and of N-S/E-W scale ratio:

By what factor differ the map areas given to identical-size earth-regions in two specified different parts of the map? By what factor does the N-S scale exceed the E-W scale at a point, where that point is chosen in places in some region of interest, or throughout some specified percentage of the earth’s surface?.

So those two kinds of distortion aren’t incomparable. They’re numerically comparable, by the factors referred to above.

But really here is the best way to compare them: Considering how little Eckert’s arctic magnification is greater than that of NGS Winkel, would you accept NGS Winkel’s humungous shape-distoriton, N-S/E-W scale disparity, over ¾ of the Earth’s surface, in trade for NGS Winkel’s slightly lower arctic magnification? …to the point where nothing between lat 49 and lat -49 resembles what it’s supposed to represent?…to the extent that ¾ of the Earth is portrayed“FUBAR”?

So, compare them in regards to whether you’d accept one kind of distortion, its magnitude on the map, to fix another distortion, in the magnitude that the map has it.. That will tell you which is worse.

2. When
something is described as “ugly”, whether a sculpture, a car, or forms in a painting, how often is “ugly” used to refer to its size rather than its form or shape?

This analogy is inapt. It’s not about absolute size. It’s about disproportion, and certainly people notice disproportion and even think it grotesque in some contexts.

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

But there’s a natural expectation for things to resemble, be shaped like, what they’re supposed to represent. (In Hammer-Aitoff and Quartic, shapes are distorted near the outer meridian, even more-so at high latitudes—but (especially in Hammer-Aitoff) that just enhances those maps’ realistic globular appearance, and doesn’t look displeasing, wrong, unaesthetic or unrealistic.)

If things don’t resemble what they’re supposed to represent (or at least how they might look on a globe, at its periphery as seen in perspective by you), then your map (like NGS Winkel) looks more like some funny-looking cartogram.

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.

4.. Do you think that, in Eckert III, the magnification of Greenland is achieved at the expense of Ecuador’s fair size or shape?

No. The magnification of Greenland happens because the meridians don’t converge quickly enough toward the poles.

Right. So, no harm done. Different magnifications in different regions is natural and expectable. …and of course necessary if we want conformality. …but as mentioned, Hammer-Aitoff and Quartic get accurate areal relation without NGS Winkel’s ridiculous wrong-looking-ness of shape.

But those four statements I made, regarding shape vs area, are probably not really so easy to justify disagreement with.

I already have, repeatedly. You are willing to salve your preferences by making sacrifices that other people find more objectionable than the problems you ameliorate.

…_some_ other people.

I haven’t taken a poll (though I soon will do so). So NGS Winkel only magnifies the Arctic by a factor of 1.9 instead of 2.58—at the cost of grossly, grotesquely, distorting ¾ of the Earth’s surface by a humungous disparity of N-S/E-W scales.

But, again, this isn’t intended as social protest. It’s just arguments intended to ask you if you really like what I describe in the paragraph before this one, for example.

That’s fine. I repeat, you need real data for your claims that your preferences are shared by most or even many people

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? Right. Why aren’t they? Because no one wants them. Maybe people would buy Oxford Winkel (unexpanded), Times Winkel, Hammer-Aitoff, Quartic, or Eckert III.

Other than that, I don’t have data, but I haven’t heard of any such data provided by advocates of NGS Winkel or Robinson. As I said, I will soon do a poll, but of course I won’t be able to get poll-responses from a large sample of people, and it won’t be a scientific poll with statistically-reliable conclusions.

, and even with such data, given that those preferences are mere prejudices…

Yeah? How is that determined? “Prejudice” implies that those preferences result from pre-judging without looking at some information that is necessary for a “better” preference—information that is known to those with better preferences. But preferences are subjective. Though someone can prefer based on objective facts, like numerical facts, there’s no need for a preference to be of that kind. Such a purely subjective preference is not irresponsible or wrong.

As I said, to each their own, as regards what looks best to a person. And different people can (not wrongly) have different preferences regarding what properties they value more. If many cartographers believe that some preferences are “prejudices”, or wrong, then I suggest that cartographers are wrong about that.

…, 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.

I thought we were talking about aesthetics, not education.

I didn’t. I thought we were talking about how the Robinson projection’s interpolation method was defined, and in the absence of any standard method, whether the Robinson projection could even be called a map.

Yes, that was my first question in this topic. I understand, and don’t disagree with, your statement that even a scrawl on a restaurant napkin can be called a “map”. In that sense, sure Robinson’s is a map too. So are purely schematic maps, and cartograms.

But, other than Robinson’s, all atlas maps currently frequently encountered meet a higher standard for being maps: They’re specified mappings of lat/lon to X/Y.

(As I was saying, Azimuthal Equal Area is published without specification of the parameters needed for lat/lon to X/Y conversion, but someone could find them, if they really wanted to, by solving a system of equations.)

You seem to have abandoned that particular conversation.

We’d both said all that we had to say about it—except that, above, I’ve added a bit.

Then there was some digression about why NGS chose Winkel tripel, at which point you began making many claims about æsthetics, the value of which claims I disputed. I’m not interested in debating what’s prettier; while I have my own preferences, I would no more debate those than I’d debate food preferences.

Ok, but I just point out some considerations and comparisons regarding the aesthetics and usefulness of NGS Winkel, Robinson, and Eckert III. (and some others, including Oxford Winkel, Times Winkel, Aitoff-Hammer, Quartic, and Eckert IV--Eckert IV doesn’t have significantly worse (or even as bad) equatorial N-S/E-W scale disparity, and it has genuine equal-area, instead of NGS Winkel’s 1.9 arctic magnification.)

So cartographers want to educate people’s preferences to match the current preferences/fashions of cartographers?

Please stop raising straw men. Being acquainted with an awful lot of cartographers, I assure you there is no solidarity in preference or recommendation amongst 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. But the most likely explanation seemed to be that the publishers were all advised by the same set of cartographers, and that Robinson is the popular fashion among cartographers.

I’m not saying that Eckert III, Robinson, and NGS Winkel are the only world maps to consider. I’ve just been comparing Robinson and NGS Winkel to a similar map, Eckert III, and telling some ways in which Eckert III is better than they are. I do that because of the widespread use of NGS Winkel and Robinson. .especially Robinson. Maybe NGS Winkel is widely used only in NGS publications—which of course are found everywhere, with their NGS Winkel map.

If you call their preferences “prejudices”, that sounds elitist. So is the notion that it’s necessary to educate people so that their preferences match those of cartographers, among whom Robinson and Winkel are currently the big fashion.

Do you willfully twist what I write?

No, I sincerely answered what I thought that you meant. I apologize if I’ve misunderstood your meaning.

If everyone used Winkel tripel, then that would be inculcating a prejudice.

Not everyone does—But NGS’s maps and publications are _everywhere_, giving NGS Winkel an almost universal prominence—challenged only by Robinson’s near-complete exclusive use in world atlases.

If everyone used Robinson, then that would be inculcating a prejudice.

…but nearly everyone does :^) …when we’re referring to atlas-publishers. I call that a fiasco. Not something serious, or something to be angry about. Just one of those funny and silly things.

Or Eckert III.

Though I like Eckert III, of course it’s rarely encountered, at least nowadays.

Eckert maps used to be used more than they are now, and I feel they were a lot better than what’s usually encountered now. Eckert IV, it seems to me, used to be particularly popular in atlases. Its equatorial N-S/E-W scale disparity was justified by the goal of achieving equal-area in a high-space-efficiency map without so much shear distortion as an elliptical map.

Or Fuller dymaxion. Or Mercator. Or Briesemeister.

Briesemeister is a beauty, with its revealing portrayal of the Americas and Eurasia as one big landmass with a little split (the Atlantic), a split so recent that it isn’t very wide. …and with the map centered on that American-Eurasian landmass, showing the closeness of N. America and Eurasia across the Artic.

Or only equatorial aspects.

Sure, but I’ve been talking about them because equatorial-aspect Robinson and NGS Winkel are what we nearly always encounter.

Or any particular projection or presentation. World maps are all grossly wrong. To the extent that the public imagines a particular projection looks right, the public is wrong and ignorant.

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?

Here is what I said I didn’t notice: I said that I don’t notice that Winkel and Robinbson don’t magnify Greenland as much as Eckert III does.

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.

Let me state again these maps’ arctic magnification-factors:

Eckert III: 2.58
NGS Winkel: 1.9
Robinson: 1.75

Eckert doesn’t magnify the Arctic much more than NGS Winkel does.

(As I mentioned, I’m probably understating Winkel’s and Robinson’s arctic magnifications, due to the Simpson’s rule parabola having a maximum in the interval of area-measurement, when I measured the area of a quadrant of Robinson and NGS Winkel using only one Simpson application (one parabola) from equator to pole.)

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.

There are many answers to that. I’ve given a few above. I’ll add one or a few more, maybe repeating one in different words:

As described above, Eckert III doesn’t magnify the Arctic much more than NGS Winkel does.

Yes the Arctic and Antarctic are important. But it’s one thing to say that they’re important too, but it’s quite another thing to say that the accurate portrayal of their areas is so important as to justify the gross distortion of ¾ of the Earth’s surface via an unnecessary and gargantuan disparity of N-S and E-W scales. (…which is what NGS Winkel does)

It isn’t just that population density is low in the Arctic and Antarctic. It’s also that, together, the Arctic and the Antarctic only occupy 1/12 of the Earth’s surface area. …and, as I’ve mentioned, NGS Winkel drastically distorts shapes and the relation of N-S and E-W scales over ¾ of the Earth’s surface.

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

Isn’t that an extreme denunciation? Sure all flat world maps distort, but when that distortion is specified and known, how bad is it really? As a cartographer, you well know of the many reasons why flat world maps are useful, valuable, and even necessary.

, but like other reasonable projections, it’ll [Eckert III] do once in a while for some purposes.

Then let’s have it in atlases and magazines once in a while, instead of just Robinson and NGS Winkel. I suggest that it’s a very good alternative to them.

What is noticeable at a glance is that Eckert III, Robinson, and Winkel all seem to show Greenland very roughly twice the size of North South-America. That’s just at a brief first glance.

I cannot figure out what you’re saying here

I’d defined “North South-America” as the part of South America that’s north of the Equator.

, 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?

(And, as I said, my measurement of 1.9 for NGS Winkel is likely an underestimate, due to the Simpson’s rule parabola having a maximum in the interval of area-measurement (a quadrant of the NGS Winkel map, and of the Robinson map) ).

, 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?

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?

, but your repeated claims that your preferences are norms of some kind just don’t mean anything to me.

I hereby back away from such claims. But the buying public seem to agree with me about the desirability of NGS Winkel and Robinson, as described above in this reply.

This is what you wrote. It came with no qualification, and the context suggested no such qualification, let alone anything “explicit” about currency or frequency of encounter:

I don't claim to use or encounter all map projections. But, with the exception of Robinson's, the specifics of all projections can be found, are available to the public.

I didn’t realize that I’d said it that way. I didn’t mean it the way that I said it. I meant to refer only to currently-encountered maps.

More importantly, the reason you wrote it was to defend your claim that a projection whose details you do not know cannot even be called a map.

I qualified that statement above in this reply.

However, when pressed, you then stated,


"Then what is it about? It’s a suggestion that it would be good to state a map’s projection."




Of course it would be good, but that’s not at all what were discussing. What were discussing was this claim:


"...means that Robinson's, when found in an atlas or as a published wall-map, can't accurately be called a map.
"

I meant “…by the higher standard met by all maps other than Robinson that are currently encountered in atlases, or currently frequently encountered. (a specified mapping of lat/lon to X/Y”

So please stop changing your claims.

I’d mis-worded my claim. Above in this reply I’ve qualified and clarified the statement.

That’s interesting, and new to me, about the computational difficulties and need for numerical methods that some projections present, even when a closed-form solution is available, but not practical.

Easier to solve a small equation whose solution is writeable in closed-form, as opposed to an equation requiring Newton-Raphson

Even that does not follow. Would you really rather solve a quartic equation in closed form rather than via Newton-Raphson? Or even a cubic, for that matter?

Yes, I spoke too soon when I said that. I’ve never analytically solved a cubic or quartic equation, and would use a numerical method instead.

Michael Ossipoff

(I’m using my full name largely so that anyone searching for my name at a search engine might find these posts)