Mapthematics Forums

Triggered by two German articles which obviously misunderstood some things about Flat Maps that improve on the Winkel Tripel by Goldberg, Gott and Vanderbei, I decided to write a blogpost regarding this matter.
In preparation, I stumbled across
two blogposts by Matthew Edney who harshly criticizes Goldberg, Gott and Vanderbei’s paper as well as the Goldberg-Gott projection comparison metric in general.

While I share Edney’s opinion that it’s, ummmm, rather bold to talk of a “radically new class” of maps when, as the authors do point out themselves, “many old atlases featured two circular maps of the two hemispheres appearing side-by-side” and that it’s highly questionable how they come of with the boundary cut error of 0 for their map, I don’t feel qualified to judge the Goldberg-Gott metric.

So what do you think about all this?

Kind regards,
Tobias

Hello Tobias.

Glad you asked. Kind of. Sensationalized claims from eminent scholars are uncomfortable to deal with. I’m very pleased Matthew Edney took it on instead of my doing so.

I mostly agree with Edney’s critiques. The salient point is, “You can’t call this the best map projection because it’s not a map. You’re comparing apples to oranges. A map is one plane that we view. Your thing is two planes. You’re changing the definition of a map and then claiming you have the best map. Not a valid comparison.”

I think Edney makes a little too much of the “boundary cuts” vagueness. Boundary cuts (interruptions) are critical to characterizing lower limits on projection distortion, so it’s not fair to glibly point out Goldberg et al do not explain why boundary cuts are significant to “error”. Taken literally, Edney’s criticism here amounts to “You didn’t precisely explain why this is relevant, even though we all know they are,” which is, at worst, an omission, not a failure of the theory. But Edney seems to be skeptical, or trying to cast doubt on, the very proposition that the boundary cuts, or their details, are even relevant. That view would be wrong.

The rest of Edney’s critiques look right to me.

— daan

daan wrote: ↑Sat Apr 17, 2021 2:53 pm The salient point is, “You can’t call this the best map projection because it’s not a map. You’re comparing apples to oranges. A map is one plane that we view. Your thing is two planes. You’re changing the definition of a map and then claiming you have the best map. Not a valid comparison.”

Phew! I’m glad that you say that – because that’s what I was thinking.

daan wrote: I think Edney makes a little too much of the “boundary cuts” vagueness. (…)
The rest of Edney’s critiques look right to me.

Alright, thank you!

By the way, I actually like the idea of a double-sided azimuthal map – I liked it so much that I think I will make one for my own personal use, that’s why I was asking about the “rotation” field in the typesetting databases recently. However, meanwhile I think that I’ll just use Geocart’s raster reprojection. That will of course distort the labels by according to my tests they will still be legible.

However, I will not think that it’s the most accurate map there is…

Kind regards,
Tobias

Projections onto surfaces other than the plane can be interesting from a theoretical perspective, but probably have little practical use. Especially when the projection surface cannot be embedded in three-dimensional space in a smooth way (continuously-differentiable / no folds), as is the case for this "two disks" thingy.

I follow a simple rule that anyone who claims to have found "the best map projection" is talking nonsense and should be ignored. Anyone who claims to have found "the best map projection for a specific purpose" may actually be worth listening to. It may even be a common purpose, but unless you specify what your map is supposed to be used for (or if you claim it's suitable for all uses, both common and rare), you clearly don't know what you're talking about.

Also, the concept of two-hemisphere azimuthal maps is nothing new. People have already been using them quite a lot for applications where the discontinuity between the hemispheres isn't an issue. And this "new" projection is just the azimuthal equidistant? Seriously?

If you do happen to be impressed by the notion of gluing several projections to each other, though, there's an idea I've toyed with once of making a true "cylindrical" projection: use a cylindrical projection for the tropics and two azimuthal projections for the polar regions, printing them on the surface of a solid cylinder. If you choose the cutoff parallel right, you can arrange this so that this projection is fully-continuous as a theoretical 2-manifold, though its practical representation in 3-space as a physical cylinder still has corners (but at least they're only 90 degrees rather than 180 degrees as in a two-hemisphere azimuthal projection). This idea was born from the observation that even though they're problematic for displaying the full world, even very simple cylindrical and azimuthal projections already do a surprisingly good job displaying half of the world (the 30 south to 30 north region for cylindrical projections, or one hemisphere for azimuthal projections), with distortion across-the-board being satisfyingly low regardless of whether you use an equal-area, equidistant, or conformal projection. From there it was an easy thought to try combining the two, creating a three-dimensional shape vaguely similar in dimensions to a globe but much easier to make.

Of course, if you're doing this kind of thing then you might go all-out and try for polyhedral maps, though those have significantly harder math.

Another pretty-useless corollary of the "hemispherical azimuthal projections have satisfyingly low distortion" observation is that a map of the projective plane, which already consists of only one hemisphere (antipodal points on the sphere are identified with each other), could actually be pretty accurate across-the-board. Too bad we don't have any projective-plane planets...

Milo wrote: ↑Tue Apr 20, 2021 3:51 am Also, the concept of two-hemisphere azimuthal maps is nothing new. People have already been using them quite a lot for applications where the discontinuity between the hemispheres isn't an issue. And this "new" projection is just the azimuthal equidistant? Seriously?

To be fair, Goldberg, Gott and Vanderbei do mention that two-hemisphere azimuthal maps have been used before (“Many old atlases featured two circular maps of the two hemispheres appearing side-by-side”), and never claim that they’ve developed a new projection, although the latter has been misunderstood by at least two German articles I’ve read.

Kind regards,
Tobias

Hello everyone,

This is my first post in this forum, although I've been corresponding with Tobias by email for over a year now.

While their double-sided maps don't have any "boundary cuts", they do have a "boundary fold", and in my opinion, it's questionable whether a direct comparison with maps without any boundary folds is meaningful. An ant walking over the rim of one of their double-sided maps would arrive at the right place on the other side of the map, unlike with a boundary cut, but the rim would still present an obstacle to the ant that isn't present on the globe, so in my opinion, it's still a type of distortion.

Nevertheless, I think their double-sided maps are an interesting, although perhaps not entirely original, idea.

While I agree with Matthew Edney that some of the claims in their new paper are a bit sensationalist, I also take issue with some of his criticism of their 2007 paper that first introduced flexion and skewness. In this one of his two blogposts on the subject, he presents five of his objections in a numbered list, which makes it very easy to respond to them:

1. I also question the desirability of including "distance error" in the metric, but I don't see why it's wrong to do so if you want to. After all, it's clearly a type of distortion.

2. As daan has already mentioned, everyone already knows why boundary cuts are a significant type of distortion.

4. The different types of distortion are normalized to their values for the plate carrée, which is a type of weighting. They also take care to emphasize that this is just an example of how the different types of distortion could be combined: "This approach is certainly not unique. One may take issue with the weighting of the individual parameters, the domain over which they are applied (the whole earth, as opposed to continents only, for example), or even how the parameters are computed, we present it as a simple example of how our results may be combined with previous studies of map projections."

5. Flexion and skewness are not derivatives of isotropy (angular) and areal distortions -- the isotropy and areal distortions can be calculated from the projection metric, while flexion and skewness can be calculated from the derivative (in the mathematical sense of differentiation) of the projection metric. Note that since flexion and skewness have units of 1/distance, normalizing them to some value as Goldberg and Gott do is the only way that they can be meaningfully combined with the dimensionless quantities that characterize other types of distortion.

You might have noticed that I skipped 3. That's because I agree with Matthew Edney that some more explanation would have been welcome here.

Hello, PeteD! Nice to have you here, and thank you for your reply!

Milo wrote: ↑Tue Apr 20, 2021 3:51 am I follow a simple rule that anyone who claims to have found "the best map projection" is talking nonsense and should be ignored. Anyone who claims to have found "the best map projection for a specific purpose" may actually be worth listening to. It may even be a common purpose, but unless you specify what your map is supposed to be used for (or if you claim it's suitable for all uses, both common and rare), you clearly don't know what you're talking about.

Glad to have a partner in spreading this truth!

use a cylindrical projection for the tropics and two azimuthal projections for the polar regions, printing them on the surface of a solid cylinder. If you choose the cutoff parallel right, you can arrange this so that this projection is fully-continuous as a theoretical 2-manifold, though its practical representation in 3-space as a physical cylinder still has corners (but at least they're only 90 degrees rather than 180 degrees as in a two-hemisphere azimuthal projection). This idea was born from the observation that even though they're problematic for displaying the full world, even very simple cylindrical and azimuthal projections already do a surprisingly good job displaying half of the world (the 30 south to 30 north region for cylindrical projections, or one hemisphere for azimuthal projections), with distortion across-the-board being satisfyingly low regardless of whether you use an equal-area, equidistant, or conformal projection. From there it was an easy thought to try combining the two, creating a three-dimensional shape vaguely similar in dimensions to a globe but much easier to make.

I can only imagine the Cylinder Earth Society growing up and flourishing around these “globes”.

Another pretty-useless corollary of the "hemispherical azimuthal projections have satisfyingly low distortion" observation is that a map of the projective plane, which already consists of only one hemisphere (antipodal points on the sphere are identified with each other), could actually be pretty accurate across-the-board. Too bad we don't have any projective-plane planets...

Science fiction novel forthcoming… ?

— daan

Welcome to the forums, PeteD, and thanks for the insightful contribution.

PeteD wrote: ↑Thu Apr 22, 2021 5:59 am 1. I also question the desirability of including "distance error" in the metric, but I don't see why it's wrong to do so if you want to. After all, it's clearly a type of distortion.

I think the objection is about jumbling up distortion metrics that are not independent of each other. This is especially notable given recent work demonstrating that metrics for “distance errors” correlates closely with Jordan-Kavrayskiy-like distortion measures. If you create a dependent metric by combining other metrics (whether explicitly or through strong correlations), then your weights for all metrics involved are not directly the values you state. This is because the values of the two independent metrics (e.g. angular deformation and flation (areal inflation/deflation)) that combine to form a constant dependent distortion value can vary freely. That is, in some hypothetical error metric called “errnie”, let’s say the measure at a point is found by adding the arctan of twice the angular deformation to the areal inflation at that point. (Vaguely something like a distance error metric.) An infinite number of combinations of angular deformation and flation result in the same value for errnie. Therefore, if you weight errnie by a constant factor, and independently weight angular deformation and flation by each their own constant factors, then you’ve jumbled up the relative contributions of angular deformation and flation despite having weighted them constantly.

Combinations of skewness and flexion are very distinct from combinations of angular deformation and flation (as you note below), and this shows up strongly when ranking maps’ “total distortion” via a combination of flexion and skewness versus a combination of angular deformation and flation. Therefore, it could make sense to combine the four (plus some boundary cut voodoo?), but adding in other measures that are not statistically distinct reduces the reliability and consistency of the results.

I suspect you understand all of this, and possibly just don’t agree that the dependence is a problem because it changes the weights of the independent metrics in some dynamic way that is, or could be, considered desirable.

4. The different types of distortion are normalized to their values for the plate carrée, which is a type of weighting. They also take care to emphasize that this is just an example of how the different types of distortion could be combined: "This approach is certainly not unique. One may take issue with the weighting of the individual parameters, the domain over which they are applied (the whole earth, as opposed to continents only, for example), or even how the parameters are computed, we present it as a simple example of how our results may be combined with previous studies of map projections."

I read his objection as more in the final line you left out: “No analysis of how that previous scholar had constructed his metric, how it succeeded, and how it failed (because why else do they present their own?)” Edney is arguing for parsimony here, which is part of responsible research: Why is this even happening?

5. Flexion and skewness are not derivatives of isotropy (angular) and areal distortions -- the isotropy and areal distortions can be calculated from the projection metric, while flexion and skewness can be calculated from the derivative (in the mathematical sense of differentiation) of the projection metric.

You are quite right. I should not have relied on memory when I said I agreed with the rest of Edney’s critique.

The other half of his critique, the social half, makes powerful points I wish were more widely grasped. I don’t think the cartographic community has served itself or the public well by staking out positions on how good or bad a projection is for a general world map. The technical details of the projection are actually nearly irrelevant, within the norms of available projections.

— daan

Any error metric that combines multiple factors is always going to be subjective, dependent on relatively-arbitrary choices of weighting. Even something logical-sounding like "weigh everything equally" is still somewhat arbitrary in reality, because as Daan points out "everything" is not necessarily independent, and because it's dependant of exactly which measures you use for the input metrics (there are multiple interchangable ways of assigning a value to angular deformation that all give the same results when you ask "which of these projections has the least angular deformation at this point?", but which give different results when you start averaging them or combining them with other values). While it's certainly possible for a map based on such a metric to end up looking aesthetically-pleasing, it's pretty hard to argue that it's truly "best" in any meaningful sense.

This is why I prefer maps that perfectly optimize one of the "obvious" metrics (i.e., being conformal, equal-area, or even equidistant is some sense) and then minimize the remaining distortion (which is generally easier to measure unambiguously once you've already greatly restricted the set of maps to compare in this manner) under that constraint. But as we saw in this thread, that doesn't always work out so great either.