Disappointing observations

[daan]

In my own experience (which of course has no statistical value at all), the only projections which are easy to explain to other people before they stop listening are those which belong to the "natural projection classes" as daan called them in the Projection Essentials Tutorial.

Sure, the equidistant projections like Equidistant Cylindrical, Equidistant Conic, and Azimuthal Equidistant; and the equidistant equal-area projections--Sinusoidal, Bonne and Stabius-Werner--are the easiest to explain the construction of.
Michael Ossipoff

None of those have anything to do with natural (or literal) projections.

— daan

[RogerOwens]

Sure, the equidistant projections like Equidistant Cylindrical, Equidistant Conic, and Azimuthal Equidistant; and the equidistant equal-area projections--Sinusoidal, Bonne and Stabius-Werner--are the easiest to explain the construction of.
Michael Ossipoff

None of those have anything to do with natural (or literal) projections.

— daan

Ok, I just thought that there was something natural about them, and that their construction is particularly briefly explained. ...whether they're really natural (in some other sense) or not.

Michael Ossipoff

[RogerOwens]

Disappointing observation:

There is no theory of optimal equal-area projections.

— daan

But I guess it isn't a problem for comparing or choosing Cylindrical Equal-Area (CEA) maps, because there's only one thing that is done to the land-portrayal (North-South/East-West scale-disproportion), and there's only one parameter.

Other than that, though, I guess there's the question of choice and definitions of the kinds of error looked at, and their relative weighting for comparison.

Someone suggested the comparison to a reference-projection, by which a projection's error of whatever kind would be expressed in terms of its value for the reference-projection. He suggested Cylindrical-Equidistant, because it's the simple projection that at least somewhat compromises between area and shape. But, for comparing equal-area projections, maybe some particular CEA would be better.

You're more familiar with this than I am, but I guess one choice for the two kinds of error could be 1) maximum scale variation (in different directions) at the point being looked at; and 2) maximum angular-error at that point.

I don't know--It seems to me that the departure from perpendicularity where graticule-lines meet is an especially important angular-error. The graticule-lines seem special, because they're what a person uses when trying to identify or find a lat/lon point on the map.

So, should that particular angular-error be added as a 3rd kind of error to count? It would give CEA built-in aadvantage. But, because the error, the departure from perpendicularity of graticule-intersections would be zero for the CEA reference-projection, it wouldn't work, with an error-weighting based on by dividing a map's error by that of the reference projection. So should graticule-intersection non-perpendicularity be left out, or brought in, in some other way? Maybe use Sinusoidal as the reference-projection.

Maybe also include the reference-map's area, divided by its equator-length (or its square)--divided by that quotient for the map being evaluated.

As I mentioned once before, a highly-regarded cartographer mentioned in a book that CEA has the least angular distortion of any equal-area world map. I don't remember his name.

The other matter is about how, for a particular kind of error, the errors at the various points looked at are aggregated.

RMS might be best for predicting how good the map will look. (...but maybe not, if the region with the most error, which therefore is most noticed by RMS, is a region that's less noticed by humans.)

But, for practical use purposes, wouldn't a simple mean of the error at those points be more useful. ...because it tells what the expected error is?

Michael Ossipoff

[daan]

There is an extensive literature on “optimal” equal-area maps, but whatever scheme someone proposes, the scheme comes with arbitrary elements or assumptions. This is in contrast with conformal maps, where there is a simple criterion that determines what conformal map is optimal for the region in question.

Someone suggested the comparison to a reference-projection, by which a projection's error of whatever kind would be expressed in terms of its value for the reference-projection.

I don’t know who “someone” is or what value a “reference projection” would be. The proper reference projection is the globe.

You're more familiar with this than I am, but I guess one choice for the two kinds of error could be 1) maximum scale variation (in different directions) at the point being looked at; and 2) maximum angular-error at that point.

There is no difference between your two measures.

I don't know--It seems to me that the departure from perpendicularity where graticule-lines meet is an especially important angular-error. The graticule-lines seem special, because they're what a person uses when trying to identify or find a lat/lon point on the map.

The problem, as I pose it, is independent of human biases such as a desire for perpendicular graticule elements. Such a criterion has nothing to do with how abused by the projection the underlying terrain is.

As I mentioned once before, a highly-regarded cartographer mentioned in a book that CEA has the least angular distortion of any equal-area world map. I don't remember his name.

As I mentioned before, this is false. None of the myriad papers exploring ways to construct or measure an optimal equal-area map have concluded that a cylindric equal-area projection is optimal, whether using RMS, “mean error” (as you put it), Q-value, Jordan-Kavraiskiy criterion, Airy-Kavraiskiy criterion, Golberg & Gott’s skewness, Canters’s scale correction to either of the -Kavraiskiy criteria, finite distance error minimization by any number of different aggregation methods… or anything else.

The only exception is Goldbert & Gott’s flexion, in which the cylindric equal-area by the three different standard parallels tested comes out ahead of the other equal-area projections tested. However, this is not meaningful on its own because flexion is paired with skewness in Goldbert & Gott’s scheme as two distinct measures. By skewness, the cylindric equal-area fared dead last. Therefore, if you combine the two measures into one somehow, cylindric equal-area will lose. Also, by flexion’s measure, it is the Lambert form (standard parallel is equator) of cylindrical equal-area that tests as least distorted, and yet Lambert is the most distorted by every other measure ever used. This strongly indicates that flexion on its own is not a useful measure of optimality. Hence, by every metric of optimality ever encountered in the literature, cylindric equal-area projections of whatever standard parallel carry higher angular deformation in aggregate than other choices do.

I already debunked this notion of perpendicular parallels and meridians having any significance in angular distortion measure in my posting on shear. Now I have debunked it by every study ever done on equal-area optimality. I wonder how much longer failed musings will continue to be aired.

The other matter is about how, for a particular kind of error, the errors at the various points looked at are aggregated.

This is entirely what it is about.

— daan

[Atarimaster]

As I mentioned once before, a highly-regarded cartographer mentioned in a book that CEA has the least angular distortion of any equal-area world map. I don't remember his name.

As I mentioned before, this is false. None of the myriad papers exploring ways to construct or measure an optimal equal-area map have concluded that a cylindric equal-area projection is optimal, whether using RMS, “mean error” (as you put it), Q-value, Jordan-Kavraiskiy criterion, Airy-Kavraiskiy criterion, Golberg & Gott’s skewness, Canters’s scale correction to either of the -Kavraiskiy criteria, finite distance error minimization by any number of different aggregation methods… or anything else.

Maybe RogerOwens refers to Wagner, who mentioned in Kartographische Nachrichten (1949) that the Behrmann projection has "particularly beneficial" Durchschnittswinkelverzerrungen (literal translation "mean angular distortions", I don’t which term is used in English literature) "among a certain amount of other well-known projections".
Later in that book, when Wagner talks about Eckert IV, he says that the term Durchschnittswinkelverzerrung was introduced by Mr. Behrmann (so now you know to which kind of measurement that term refers) and that Eckert IV has the lowest value of that kind "among the better-known world map projections" and is "surpassed only by the cylindrical equal-area projection with φ0 = 30°".

So in both cases, Wagner specifically refers to well-known projections but not to all projections.
But even using Behrmann’s way to measure distortions, newer projections like Aribert Peter’s Entfernungsbezogene Weltkarte of 1978(*) and Hufnagel 12 (1989) have lower distortion values than Behrmann’s cylindrical equal-area.
It also should be noted that Hufnagel 12 was build specifically to reach low distortion values but that Mr. Hufnagel himself did not regard his No. 12 as a good projection for practical purposes.




*) The Entfernungsbezogene Weltkarte (distance-related world map) is a Wagner VII variant. The image that I linked to above is a visual approximation that I obtained by using Geocart’s generalized Wagner using the parameters
a = 2.74804, b = 1.26057, m = 0.866025, m2 = 1, n = 0.333333
I don’t know the actual parameters that Aribert Peters used.

[daan]

Many thanks, Tobias. It’s good to have a name attached to the claim. That gives us something to talk about.

“Angular distortion” is common in the English literature. I prefer “angular deformation”.

Does Wagner indicate how he aggregates the measure of distortion across the entire map? That’s the controversial part. Different methods yield different results. Any of them are (more or less) defensible.

I’m going to start a new thread about my own thoughts on the theoretics of optimal equal-area projections.

— dean

[RogerOwens]

Someone suggested the comparison to a reference-projection, by which a projection's error of whatever kind would be expressed in terms of its value for the reference-projection.

I don’t know who “someone” is

The article said that it was Laskowski, writing in 1998.

or what value a “reference projection” would be.

The proper reference projection is the globe.

Sure, but they were just talking about how compare different kinds of departures from the globe, when those measures are in different units.

I didn't mean to advocate that. What occurs to me would be to estimate the range of each distortion's variation among the popular projections, and label it as a range of 0 to 1. So all the distortions would be indicates by that 0 to 1 rating.

...or, better yet, if a person could just subjectively estimate how much each kind of distortion seems to make it difficult for him to identity or find a place on the map, then he could subjectively weight the distortions in that regard. I don't know if a person can make that judgment though.

But, if equal-area projections need only be compared by angular-distortion, and that amounts to the same thing as an aggregation of the maximum scale-variation at each point, then there'd be no need for comparing different kinds of distortion. ...except that I'd want to include map area divided by the square of equator-length as a merit-measure for equal-area maps. ...and I guess I'd compare those two merit-measures by a 0 to 1 scale representing their their max variation among the popular projections.

Finding and identifying positions on the map is what seems important to me. If the map is cylindroid, and you have a latitude-ruler, then it will be easy to find or identify positions on the map. But maps that have more angular-distorition, or are smaller, make it more difficult to judge positions accurately without ruler-measurement, and that's what I'd rate equal-area projections by.

...because one of the most important uses of equal-area is thematic maps, on which it's often necessary to judge position.

I used to prize linearity, for position determination, but a latitude-ruler allows positions to be determined on any cylindroid map. A latitude-ruler would be marked to indicate, for any y-value on the map: 1) Latitude; 2)Area-error-factor (for maps that aren't equal-area); and 3)East-West scale. It would also have a scale marked to indicate Longitude for any x-value on the map.

If the map is cylindrical, then the latitude-ruler could also have a scale marked to indicate the North-South scale for any y-value on the map. ...or maybe just the NS/EW scale disproportion. That's an advantage that a cylindrical has over a pseudocylindrical--the possibility of showing that with a latitude-ruler.

Of course both sides of the ruler would be needed, to have all those scales.

There is no difference between your two measures.

That's a good thing.

I was suspecting that, even when posting my message. I should have known it, because a conformal projection, by equalizing the scale in all directions at every point, gets rid of angular errors and give accurate small shapes.

So, for an equal-area map, would one just aggregate the maximum scale-variation at lots of points? Would that be the measure of shape-accuracy?

I don't know--It seems to me that the departure from perpendicularity where graticule-lines meet is an especially important angular-error. The graticule-lines seem special, because they're what a person uses when trying to identify or find a lat/lon point on the map.

The problem, as I pose it, is independent of human biases such as a desire for perpendicular graticule elements. Such a criterion has nothing to do with how abused by the projection the underlying terrain is.

Yes, I was beginning to doubt what I'd said as soon as I posted it. Arguably the angular-error measured between the graticule-lines at their intersection is the least deceptive angular error, because the graticule-lines make it visible and obvious.

The other matter is about how, for a particular kind of error, the errors at the various points looked at are aggregated.

This is entirely what it is about.

Ok--Is that because equal-area maps' shape-accuracy can be measured by just finding and aggregating the maximum variation of scale in different directions at each point looked at?

So, if one is only interested in shape-accuracy, there's only one distortion-measure to look at, and so there's no need to weight distortion measures?

Michael Ossipoff

[RogerOwens]

Maybe RogerOwens refers to Wagner, who mentioned in Kartographische Nachrichten (1949) that the Behrmann projection has "particularly beneficial" Durchschnittswinkelverzerrungen (literal translation "mean angular distortions", I don’t which term is used in English literature) "among a certain amount of other well-known projections".

[...]

Later in that book, when Wagner talks about Eckert IV, he says that the term Durchschnittswinkelverzerrung was introduced by Mr. Behrmann (so now you know to which kind of measurement that term refers) and that Eckert IV has the lowest value of that kind "among the better-known world map projections" and is "surpassed only by the cylindrical equal-area projection with φ0 = 30°".

Of all the CEA maps, the Behrmann looks best, to me.

I also like the fact that half of the Earth's surface is closer to the equator than lat 30 or -30, and half is farther from the equator. ...making lat 30 a good compromise for standard-parallel.

Michael Ossipoff

[daan]

What occurs to me would be to estimate the range of each distortion's variation among the popular projections, and label it as a range of 0 to 1. So all the distortions would be indicates by that 0 to 1 rating.

In which case, anything with a pole line would have a rating of “1”, if 1 indicates infinite distortion.

...or, better yet, if a person could just subjectively estimate how much each kind of distortion seems to make it difficult for him to identity or find a place on the map, then he could subjectively weight the distortions in that regard.

That’s a far departure from a theory of optimality.

I'd want to include map area divided by the square of equator-length as a merit-measure for equal-area maps.

I have no idea what motivates this. It seems… bizarre.

So, for an equal-area map, would one just aggregate the maximum scale-variation at lots of points? Would that be the measure of shape-accuracy?

That’s what’s at issue in a theory of optimality: How to aggregate without just being arbitrary. I don’t think “a lot of points” is useful in a theory of optimality. It has to be all points. “A lot of points” is what’s used in the many ad hoc measures of regional or global distortion.

Arguably the angular-error measured between the graticule-lines at their intersection is the least deceptive angular error, because the graticule-lines make it visible and obvious.

I cannot follow this. I have noted, over and over, that the angle of intersection of latitude and longitude says nothing whatsoever about the local angular deformation.

Ok--Is that because equal-area maps' shape-accuracy can be measured by just finding and aggregating the maximum variation of scale in different directions at each point looked at?

No, it’s because, as I noted above, without a theory of optimality, how to aggregate is merely a matter of arbitrary choice. How you aggregate changes the rankings of the projections involved.

So, if one is only interested in shape-accuracy, there's only one distortion-measure to look at, and so there's no need to weight distortion measures?

I’m not sure what you’re saying here, but shape accuracy is all that matters in an equal-area map. By virtue of being equal-area, all disproportion of area is gone, leaving only angular deformation, which is the same as shape inaccuracy. There is a third kind of “distortion” that gets largely ignored in the literature, which is disruption of contiguity. The more interruptions you permit, the lower the distortion can be within the contiguous section. It is not fair to compare a highly interrupted map to one that is only split along one meridian. However, presuming you interrupt each projection the same, then, in an equal-area map, “shape accuracy” is the only concern in a theory of optimality.

I would suggest starting a different thread if you intend to discuss this further. “Disappointing observations” really is not apt.

— daan

[quadibloc]

(Not to mention that in some parts of the world, the maps aren’t even centered to the Greenwich meridian.)

Oh, you mean like this

(Russia)
or like this

(China)
or like this

(The United States)
...all, right, I can relate.
I thought you meant something else. I remember, back in the sixties, as a young lad, seeing the "Atlas Jeune Afrique" in the local public library, and being quite shocked that the maps in it had no graticule. I assumed, perhaps incorrectly, that some countries in Africa found the Greenwich meridian politically objectionable.
Despite the prime meridians of Paris and Ferro and even Washington D.C. going by the boards, and most of the world having an international agreement on accepting the Greenwich meridian, is there any place on Earth still using another one?