Pareto-Optimality of Maps: What Metrics are Best?

[Green5]

I understand that there is no such thing as a perfect 2D projection of a sphere, but perhaps it will be useful to find out which are Pareto-optimal: projections which are not strictly inferior to any other projection, based on some metrics. And in the process identify strictly inferior projections.

The metrics used by Goldberg and Gott in this paper (https://www.physics.drexel.edu/~goldber ... g_gott.pdf) seem useful for such a purpose, and of the 27 maps that they listed in the metric table, 7 are not Pareto-optimal: Eckert VI, Gall-Peters, Hammer, Lambert Conic, Mercator, Mollweide, and Sinusoidal.

(Yes I know they recently applied their metrics to their double-sided disk map to claim better Distance and Boundary Cuts errors, which is technically correct but disingenuous to compare with 2D maps with 1 surface... But I think their metrics are useful for thinking about a 6D Pareto frontier)

My question is, do you have any suggestions for other metrics to be used, or any criticism of their metrics? Of their metrics, I think Isotropy, Area and Boundary Cuts are pretty solid, not sure about the other 3. I also see some conversations about metrics on this forum, maybe those can be used.

[Milo]

There are many debatable or situational benefits that you won't always care about, but that can still make a projection worthwhile in some specific cases.

In particular, there are two azimuthal projections that are clearly non-optimal by the most common metrics, the orthographic projection (which has even worse angle distortion than the azimuthal equal-area projection) and the gnomonic projection (which has even worse area distortion than the stereographic projection), which nonetheless have unique properties not shared with any other projection that make me consider them among the most important projections.

The metrics used by Goldberg and Gott in this paper (https://www.physics.drexel.edu/~goldber ... g_gott.pdf) seem useful for such a purpose,

I'm not sure what metrics they're even using? The names they give in the table are "isotropy", "area", "flexion", "skewness", "distances", and "boundary cuts", but the only ones that they have actual sections defining are flexion and skewness, and even then they're confusing about it (the section defining flexion claims that the gnomonic projection has flexion zero, but the table gives the flexion of the gnomonic cube as 0.12, low but nonzero - do the singularities somehow cause it to have higher overall flexion despite those singularities still being a measure-zero set?). They appear to be using "isotropy" to mean "angle distortion" (based on the fact that conformal projections have a value of zero), but for both angle and area distortions, it's unclear to me what exact metric they're using (note, for example, this discussion on metrics of area distortion, and how one of the more commonly-used ones, normalized flation, is of questionable value). Even the metrics they do explain are explained in a somewhat wordy fashion that's difficult for me to parse.

Yes I know they recently applied their metrics to their double-sided disk map to claim better Distance and Boundary Cuts errors, which is technically correct but disingenuous to compare with 2D maps with 1 surface...

I think that any distance-based metric is meaningless unless it addresses boundary cuts somehow. Such as by allowing you to measure the distance from point A to the boundary, then from the corresponding boundary to point B, instead of measuring the "on-page" distance of points near the boundary cut.

There is some advantage to having boundary cuts that are "easy to work with": a cylindrical projection can be scrolled seamlessly, and certain polyhedral projections (square dihedral, triangular dihetral, tetrahedral) can tile the plane (and even the ones that can't still have convenient straight-line boundary cuts), making it easier to treat the boundary cuts as if they're not there - provided, of course, that you're working in a context where such scrolling is actually possible, otherwise there's no point.

[PeteD]

Of their metrics, I think Isotropy, Area and Boundary Cuts are pretty solid, not sure about the other 3.

I agree. However, I think that combining their boundary cut measure with areal distortion and isotropy, i.e. angular distortion, into a single metric, as Goldberg and Gott do, isn't very meaningful.

You can combine areal and angular distortion with each other, e.g. the Airy–Kavrayskiy criterion, but even then it's not so simple since there's the question of the relative weightings of the two types of distortion. You can weight them 50:50, but optimizing for this leads to projections like the Györffy projections, whose areal distortion looks too high to most people. Goldberg and Gott normalize areal and angular distortion to the respective values for the plate carrée, and optimizing for this leads to projections that to my eyes have a good balance between areal and angular distortion, at least for projections interrupted along one meridian.

If you do the same with their boundary cut measure, i.e. normalize it to the value for the plate carrée, and then include this in the combined metric with the goal of comparing projections with different amounts of interruption, then all the highest-ranked projections will be projections interrupted along one meridian, as in Goldberg and Gott's paper, so this doesn't achieve the stated goal of comparing projections with different amounts of interruption.

Alternatively, if you choose the azimuthal equidistant as the reference projection to which the distortion values of other projections are normalized, then any projection interrupted along anything more than just a single point will have infinite distortion according to your metric. Or if you choose the azimuthal equidistant split into two hemispheres as the reference projection, then all the highest-ranked projections will be projections interrupted along a great circle, so again this doesn't achieve the stated goal of comparing projections with different amounts of interruption.

So yes, I agree that areal and angular (or isotropy) distortion and interruptions are the three most important types of distortion. You can combine areal and angular distortion into a single metric, although you have to choose an arbitrary weighting. Normalizing distortion values to those of a reference projection is in a sense "less arbitrary" than just picking a number for the weighting. The particular reference projection used is still arbitrary, but the plate carrée as the simplest projection interrupted along one meridian is in a sense the "least arbitrary" choice for projections interrupted along one meridian. It also seems to work quite well for these projections.

On the other hand, it doesn't seem very meaningful to include Goldberg and Gott's boundary cut measure in a combined metric. Instead, it seems that it is only meaningful to compare a given projection with other projections that have the same amount of interruption.

[PeteD]

My question is, do you have any suggestions for other metrics to be used, or any criticism of their metrics?

I have some criticism of their other metrics, particularly flexion and distance errors.

Flexion is zero everywhere for the gnomonic projection ...

<tangent>

(the section defining flexion claims that the gnomonic projection has flexion zero, but the table gives the flexion of the gnomonic cube as 0.12, low but nonzero - do the singularities somehow cause it to have higher overall flexion despite those singularities still being a measure-zero set?)

Goldberg and Gott address this in their paper:

This is a particularly interesting projection since the gnomonic is locally flexion-free, but it is clear that geodesics will not trace out straight lines in the gnomonic cube cube [sic] map because they bend when they cross an edge between faces. The gnomonic cube is presented as a cross, so 5 edges are included in the map proper. Geodesics bend when they cross an edge in this laid out cross configuration.

</tangent>

Where was I? Oh yes, flexion is zero everywhere for the gnomonic projection and only for the gnomonic projection. You can therefore think of flexion as the degree to which a given projection deviates from the gnomonic. While there are specific purposes for which the gnomonic projection is useful, for anything other than those specific purposes, it's terrible. For most purposes, it is therefore not desirable to have low flexion.

Milo makes a very good point about distance errors:

I think that any distance-based metric is meaningless unless it addresses boundary cuts somehow. Such as by allowing you to measure the distance from point A to the boundary, then from the corresponding boundary to point B, instead of measuring the "on-page" distance of points near the boundary cut.

Goldberg and Gott's distance error measure penalizes projections for putting Alaska and Kamchatka on opposite sides of the map when they're relatively close on the globe. Yes, interrupting the globe near the Bering Strait is a form of distortion, but is the Gott elliptical really any better than the WInkel Tripel in this regard? The Gott elliptical will be penalized less because Alaska and Kamchatka are closer together on the map.

Gott states in another paper that minimizing distance errors leads to circular maps. This isn't normally desirable for projections interrupted along one meridian, so minimizing distance errors is also not normally desirable for these projections.

Skewness, on the other hand, isn't so bad. Low skewness correlates quite well with a good balance between areal and angular distortions. However, if we already have areal and angular distortion in our combined metric, and if skewness is valuable only to the extent that it correlates with areal and angular distortion, then there's not much point in adding skewness to the metric.

The case could be made for using skewness instead of a combination of areal and angular distortion. This would eliminate the need to choose an arbitrary weighting. However, it has a few drawbacks:

1. It's more computationally intensive to calculate than areal and angular distortion.
2. It gives nonsensically high values for projections that are interpolated from values in a table rather than defined by a formula, and the particular values obtained are strongly dependent on the particular interpolation method used.
3. Optimizing for skewness produces projections that are too tall and narrow (but that otherwise look quite good).

[PeteD]

for both angle and area distortions, it's unclear to me what exact metric they're using

Their areal distortion measure A is identical to the E_p under sections 1 and 3 of this previous post of mine. Their angular distortion (or "isotropy") measure I is defined analogously but obviously without doing anything to make it scale-invariant:

I² = E_a² = 1/(4 π) ∫∫ ε_a² cos ϕ dϕ dλ,

where ε_a = ln a/b.

[Green5]

On the other hand, it doesn't seem very meaningful to include Goldberg and Gott's boundary cut measure in a combined metric. Instead, it seems that it is only meaningful to compare a given projection with other projections that have the same amount of interruption.

My objective is not combining all metrics into some weighted sum of each like in their paper, its to create a Pareto frontier. So it doesn't really matter how each metric scales relative to other metrics, all that matters is whether a map is strictly inferior to any other map in all metrics or not. That will allow doing what you describe, like fixing boundary cut at some value and comparing the other metrics.

[Milo]

(the section defining flexion claims that the gnomonic projection has flexion zero, but the table gives the flexion of the gnomonic cube as 0.12, low but nonzero - do the singularities somehow cause it to have higher overall flexion despite those singularities still being a measure-zero set?)

Goldberg and Gott address this in their paper:

This is a particularly interesting projection since the gnomonic is locally flexion-free, but it is clear that geodesics will not trace out straight lines in the gnomonic cube cube [sic] map because they bend when they cross an edge between faces. The gnomonic cube is presented as a cross, so 5 edges are included in the map proper. Geodesics bend when they cross an edge in this laid out cross configuration.

Yes, I realize that. (Well, not that is was mentioned in the paper, I only skimmed it.) The gnomonic cube is not flexion-free everywhere, but it is flexion-free almost everywhere (in the mathematically-rigorous sense of everywhere except for a set of measure zero), like I said. I would expect that to integrate to still zero average flexion over the whole map. But maybe it has infinite flexion wherever it isn't zero, and you get weird infinity-times-zero shenanigans?

Oh yes, flexion is zero everywhere for the gnomonic projection and only for the gnomonic projection.

A technicality: flexion is also zero (or should be, since geodesics are still straight lines) for affine transformations of the gnomonic projection, which is how the two-point azimuthal projection is constructed.

While there are specific purposes for which the gnomonic projection is useful, for anything other than those specific purposes, it's terrible. For most purposes, it is therefore not desirable to have low flexion.

Worth noting is that almost none of the listed maps score "well" at flexion. The gnomonic cube and the two-hemisphere stereographic projection are the only ones that have a value lower than 0.5. Which of course is meaningless without knowing what scale the metric is on, but the highest value of any listed projection is 1.0 (for all uninterrupted azimuthal projections that aren't gnomonic, regardless of how similar or different they otherwise look to the gnomonic projection!?) So we can't even judge low-flexion maps other than the gnomonic projection, because nobody uses them!

The Winkel Tripel projection has values below 0.5 in every category except flexion, where it has a middle-of-the-road value, not the best or the worst. (It has the lowest skewness of any non-two-hemisphere projection. Its distance error of 0.374/0.39 is less impressive when you realize that only the stereographic projection scores worse than 0.46 in that category.)

Gott states in another paper that minimizing distance errors leads to circular maps. This isn't normally desirable for projections interrupted along one meridian, so minimizing distance errors is also not normally desirable for these projections.

Well, the Lagrange projection is a pretty good meridian-interrupted map that happens to be circular, but it actually scores rather poorly by the distance distortion metric that the paper uses, with only five projections being worse (stereographic, equirectangular, Lambert conic, Mercator, Miller). It is, however, Pareto-optimal due to being the best conformal projection in the list. It's at least not Pareto-dominated by any non-conformal projection (having zero isotropy error), and it Pareto-dominates two of the other conformal projections (Lambert conic and Mercator - note that the latter also dominates the former), and almost dominates the the stereographic projection (only being worse at boundary cuts).

I'm somewhat confused by how the table even has concrete values for the Lambert conic projection. Shouldn't that depend on your choice of standard parallels?

My objective is not combining all metrics into some weighted sum of each like in their paper, its to create a Pareto frontier.

I think the same argument applies here about skewness not being particularly useful when you're already tracking angle distortion and area distortion separately. Compromise projections with acceptably-low but nonzero area and angle distortions would already appear as part of the Pareto front simply from looking at those metrics, so you don't need another metric to track those specifically unless you're actually trying to define some concept of "best compromise".

Though it's not exactly the same. Notably, the polyconic projection has lower skewness than some other projections such as Eckert IV, Eckert VI, equirectangular, Hammer-Wagner, and Mollweide, despite being worse than them at both isotropy and area. Likewise, the Breisemeister projection has the lowest skewness error of any non-two-hemisphere equal-area projection, but does not have the lowest isotropy error.

The interesting property of the polyconic projection is that all parallels have the correct shape. However, this is not true of circles in general, only parallels.

[PeteD]

A technicality: flexion is also zero (or should be, since geodesics are still straight lines) for affine transformations of the gnomonic projection, which is how the two-point azimuthal projection is constructed.

That hadn't occurred to me. Thanks for pointing that out.

I'm somewhat confused by how the table even has concrete values for the Lambert conic projection. Shouldn't that depend on your choice of standard parallels?

Yes. They use a single standard parallel of arcsin 1/3 = 19.5°.

[PeteD]

My objective is not combining all metrics into some weighted sum of each like in their paper, its to create a Pareto frontier.

In that case, I would definitely use areal and angular distortion and interruptions, as you said. You could consider using skewness as well, but bear in mind Milo's comment:

I think the same argument applies here about skewness not being particularly useful when you're already tracking angle distortion and area distortion separately.

I definitely wouldn't use flexion or distance errors.

[PeteD]

Though it's not exactly the same.

Absolutely. I only said it correlates quite well.

Notably, the polyconic projection has lower skewness than some other projections such as Eckert IV, Eckert VI, equirectangular, Hammer-Wagner, and Mollweide, despite being worse than them at both isotropy and area. Likewise, the Breisemeister projection has the lowest skewness error of any non-two-hemisphere equal-area projection, but does not have the lowest isotropy error.

This ties in with my previous point that skewness favours tall and narrow projections. The polyconic is taller than Eckert IV and VI, equirectangular (specifically the plate carrée), Hammer-Wagner and Mollweide. If you rescaled all these projections to have correct area, the polyconic would also be the narrowest. Likewise, the Briesemeister is taller and narrower than the equal-area projections with lower angular distortion: Eckert IV and VI, Hammer-Wagner and Mollweide.