On distortion and optimal projections

[daan]

Conditions that indicate “optimality” in a general map projection are subject to much debate. Inevitably, you have to make choices about what’s important to measure and what distortion means over a region—and especially over the entire earth, where the problem of discontinuities also comes into play.

Distortion comes in two primary forms: deformation of angles, and variation in relative area measure. Every map has at least one of those two primary forms. Many maps have both. One might choose to measure or characterize other things, such as “scale”, distances, directions, or bearings, and then measure how well a map preserves one of them. However, ultimately, any other metric that you choose arises out of the inevitable distortion of angles, relative areas, or both. Those two metrics are irreducible.

I will carefully explain these two forms of distortion. As described by Nicolas Auguste Tissot, 1881, in Memoire sur la representation des surfaces et les projections des cartes geographiques, you can characterize the distortion at a point as the abuse an infinitesimal circle from the sphere suffers when projected onto the plane. Because the circle is infinitesimal, only two things can happen to it: It can gain or lose area, and it can get squashed into an ellipse. Here I portray the original circle on the left and an example projection on the right:

Left, unprojected; right, the effects of projection

Tissot.JPG (31.73 KiB) Viewed 6486 times

You can imagine an unlimited number of crosses in the original circle. I show four, each in a different color. After projecting, most of the crosses can end up no longer meeting at right angles, but there will always be one that does, or possibly all of them will still. Unless all of them remain at right angles, there will always be one axis with the greatest change in angle. Among those axes shown, that would be the green one. If all of the axes keep their right angles, then there is no angular deformation at that point. The ellipse generated by this mechanism is called the Tissot indicatrix.

Secondly, the area has changed in this example. The ratio of the area of the resultant ellipse to the original circle is the amount of area inflation or deflation at the point. If the area does not change, then there is no “distortion” of area at the point, regardless of how much angular deformation there might be. Across a map, these measures change point-by-point. If all the ellipses have the same area, then the map is equal-area. If all of the ellipses are actually circles, and therefore show no angular deformation, then the map is conformal. A map cannot be both conformal and equal-area. Even so, specific points, lines, or paths can be without distortion as long as the points do not form a region, however small.

Here is a map showing the Tissot indicatrix for an entire map, spaced at 15° intervals:

Tissot indicatrices of varying sizes and varying deformation

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This map is neither equal-area nor conformal: Its Tissot indicatrices vary in area and most of them are ellipses.

By contrast, here is an equal-area map’s Tissot indicatrices:

Tissot indicatrices of identical area and varying deformation

Hammer tissot.jpg (562.21 KiB) Viewed 3577 times

All ellipses have the same area, but they show varying amounts of angular deformation.

And this is a conformal map:

Tissot indicatrices of varying area and no deformation

August tissot.jpg (1.84 MiB) Viewed 3577 times

All indicatrices are circles, but their areas vary.

“Variation in relative area measure” is cumbersome. To express the concept, the literature talks about “area distortion” or “areal distortion”. I dislike these terms because I have found them to spark misunderstandings in people’s grasp of projection distortion. Instead, I use flation, which means “area inflation or deflation”. I introduced this term in Battersby, Strebe, and Finn, 2016, Shapes on a plane: evaluating the impact of projection distortion on spatial binning.

It might seem tempting to think of angular deformation and flation as “complementary”. In a sense, they are, because together they represent everything there is to say about distortion at a point. However, they behave quite distinctly.

On the one hand, flation is relative. The scale you assign to a map is arbitrary, and so if the relative sizes of circles on a conformal map differ, who is to say which size is “correct”? You are to say… but your choice is arbitrary. All you can meaningfully talk about is the ratio of areas, and how that varies, across the map.

On the other hand, angular deformation is absolute. It does not matter what scale you assign to a map, or how you shrink or enlarge it, the ellipses still have the same proportions, and they represent the same amount of angular deformation.

Because of this very different semantic of relative flation and absolute deformation, you cannot meaningfully compare the two kinds of distortion. Let’s say, for example, that we wish to show how distortion is distributed across a map. Because the map is neither equal-area nor conformal, we can show where, and how much, angular deformation there is, and where, and how much, flation there is, using different colors. We’ll use green for flation and magenta for angular deformation. The deeper the color, the more distortion. I “bin” the colors into distinct bands, rather than show distortion in continuous tones, because it is very hard to compare color shades of regions that are distant from together, otherwise.

Angular deformation, invariant to scale changes

TsNIIGAiK angular.jpg (164.05 KiB) Viewed 3577 times

Left is the map projected at its “nominal scale”. Right is the map scaled with respect to its “nominal scale”. Notice there is no difference in the distortion pattern.

Now let’s see what happens with flation:

Angular deformation, dependent on scale

TsNIIGAiK areal.jpg (162.86 KiB) Viewed 3577 times

Left is the map projected at its “nominal scale”. Right is the map scaled with respect to its “nominal scale”. Notice the distortion pattern changes considerably. That’s because we are measuring the distortion in absolute terms against the nominal scale that we declared. In the end, the Tissot indicatrix only knows what you tell it about the size of the globe you projected from.

An interesting characteristic of projection distortion is that it is invariant against coordinate rotation. What is coordinate rotation? That’s when you grab the globe’s axis and yank the north pole to some other place. When you project using the same formulæ, you get the same outline for the map, but the contents of the map are all shifted. Neverthless, the distortion of each point in the map, and the patterns of distortion across the map, do not change. For example:

Right is the same projection, centered at 45°N

Briesemeister.jpg (209.74 KiB) Viewed 3577 times

Lo! No difference in distortion even when you move a different point to the center:

No change in distortion at any map position

Briesemeister distortion.jpg (179.13 KiB) Viewed 3577 times

Another interesting characteristic of projection distortion is that the angle formed by meridians and parallels tells you nothing about the distortion. It is true that a conformal projection always has perpendicular meridians and parallels, and this is true even under coordinate rotation. However, if the map is •not• conformal, then even if it had perpendicular meridians and parallels in one view, it would not after coordinate rotation.

So far we’ve discussed distortion as a phenomenon of points. That’s because the Tissot metric applies to individual points. What if we want to compare two different maps to ask, “Which one has less distortion?”

This is a hard problem. Many different schemes have appeared in the literature. Each has its merits. Each has its deficits. Each ranks the overall distortion of a selection of maps differently.

The problem starts out hard because distortion, even at the level of a point, consists of two independent, incomparable phenomena: angular deformation and flation. Without making arbitrary decisions about how to go about it, you cannot claim that this much angular deformation is worth that much flation. If you cannot do this at the level of a single point, how do you do it across the unlimited points of an entire map?

Nevertheless, we really seem to want answers about which projection is better, so we keep trying.

Interestingly, the question of which conformal map is best has a rigorous and beautiful answer. First some preparatory remarks. In the distortion maps I show above, you can see bands of distortion. Each band has edges, and you can think of an edge as a path. In the parlance of mathematical cartography, such a path is called an isocol or isomegeth or level curve of distortion. An isocol has constant distortion measure along its length. In the case of conformal maps, this isocol is a path of constant flation. (Originally, isocol referred only to the level curve of area measure on a conformal map, but in recent decades researchers have generalized usage to also include the path of constant flation on other maps, as well as the path of constant angular deformation on any map. Obviously, a conformal map will have no isocols of angular deformation, and an equal-area map will have no isocols of flation. A map that is neither equal-area nor conformal will have both, and on most maps, they will not coincide.)

In 1853, renowned mathematician Pafnuty Chebyshev developed a hypothesis about this problem of the optimal conformal map. First, you define what region you are interested in using a closed boundary path. When you do this, Chebyshev conjectured, the optimal conformal map for that region is the conformal map that has an isocol coincident with that boundary. In 1896, Chebyshev’s hypothesis was proved by D.A. Gravé (sometimes transliterated “Grawe”). In the century plus since, researchers have devised various optimal projections for specific regions of the world based on the Chebyshev criterion.

The reason Chebyshev’s mechanism works is because of some very deep and interesting properties of conformal maps. One of those properties is that, if you know the distortion characteristics of even just a short path in the conformal map, then you can know the entire conformal map! This arises out of something called the continuation theorem. One of its implications is that every conformal map is unique, given a bounding isocol or any other path on it. Because it is unique, you can say the map having an isocol that bounds the region of interest is the one that is optimal for that region.

But optimal in what way? The thing about maps is that distortion can go out of control easily. A map that stretches the poles into lines, for example, has infinite distortion at those points, and distortion increases monotonically toward those poles, too, and so they are not just singularities that can be ignored; the entire region nearby is grossly distorted. This means that any measure of overall distortion that includes the poles will be heavily weighted by these most-distorted regions. What Chevyshev’s criterion does is to put a concrete limit on the distortion. If you have an isocol of flation delineating a region, then at no point inside that region will the flation ever exceed that of the bounding isocol’s. The ratio of the flation of the isocol to the minimum flation within the region is the minimum it can possibly be. It is by this simple, natural, and breathtakingly lucid criterion that conformal maps can be compared against each other.

Nearly optimal for the entire US because nearly encloses the 50 states. Absolutely optimal for the enclosed northeast Pacific.

GS50.jpg (126 KiB) Viewed 6486 times

There are other ways to measure and compare flation, and by other metrics, how different projections ranking in distortion performance could change. However, with the power of Chebyshev’s criterion, little impetus for such research exists, and little has been done. There isn’t even any reason to compare. The goal is to formulate a projection with an isocol that bounds the region you care about. The closer your isocol is to your ideal region, the better you have met your goal.

What about a world map? With world maps, you have to be careful and state clearly what you mean: a world map that is only ripped at one point, like an azimuthal projection? A world map that is ripped along an entire meridian like the maps I show on this page? Something else? If you only interrupt at one point, the optimal conformal map is the stereographic, which is not very satisfying because it is infinite in extent. If you permit interruption along an entire meridian, like most world maps you see, then the optimal conformal map is the Eisenlohr.

In short, the theory of conformal projections is well developed. It draws upon the vast literature and flexibility of complex analysis.

Having disposed of this background material, I will next write a post, as time permits, about optimal equal-area maps.

[daan]

A theory of optimality


When I talk about a theory of optimality, I mean a mathematical statement that tells you how to know if you have the optimal map for the conditions that you have set. I do not mean a statement by which you can ascertain that map A is better than map B. It has to tell you that there is no better map to be made.

Obviously, it is best if the theory is proven, as Chebyshev’s was eventually. Still, even during the decades between when his conjecture was mooted and when it was proved, Chebyshev’s theorem was valuable and credible because the concepts behind it were sound. Nobody seriously doubted its veracity.

Chebyshev’s criterion stands apart from the many methods proposed to measure and compare distortion. Those are not theories. They are descriptions of statistical methods, essentially, combined with numerical minimization techniques in order to search out the best map feasible, given the computational and time resources allowed. They all come with exclusionary justifications, some more strained than others. For every reason one is good or useful, there are other reasons it’s not.

— daan

[daan]

On optimal equal-area projections


We have a theory of optimal conformal projections, a very good one as I mention above in this thread. What we do not have is a theory of optimal equal-area projections.

To be sure, we have methods of reducing angular error. These are generic minimization algorithms such as the simplex algorithm. They work fine as far as they go, but before you even start minimizing, you have to decide what it is you’re minimizing. You can’t just tell it to minimize angular error and set it off. The problem is that there are many different definitions one might use per point:

• Use the raw maximum angular deformation. This ranges from 0° – 180°. Or, 0° – 90°, depending on which definition you use.
• Use the major axis of the Tissot ellipse, which is the maximum scale. This ranges from 1 – ∞.
• Use the minor axis of the Tissot ellipse, which is the minimum scale. This ranges from 0 – 1.
• Use some average (which?) of the length of every possible projected axis in the Tissot ellipse.
• Use the logarithm (usually squared) of the major axis, typically because you’re comparing to projections that are not equal-area.
• Use Gott & Goldberg’s notion of skew and flexion—but this is two different metrics and are not intended to be aggregated.
• Or, something else.

Any of these are defensible. None are imperative. Each will give a different result and rank different projections differently.

Once you’ve settled on one—remembering that you’ve already made a hugely arbitrary choice—then you have to decide how to aggregate the points into one measure. The problem is that there are many different ways of aggregating the point measures into a total measure:

• Average them.
• Compute their root mean square.
• Weight them according to your favorite reasons that this or that local should be exempt or emphasized.
• Or, something else.

But wait. Some of those methods are not scale-invariant. If you scale the map, you will get different results. That doesn’t seem reasonable for angular deformation, which is scale-invariant. So you:

• Ignore it.
• Apply a normalization function (which?) to eliminate scale dependence.
• Maybe not use a scale-dependent point distortion metric after all—but how do you compare against projections that are not equal-area?

Since we started out just wanting to know what the best equal-area map is, let’s just forgo measures that are scale-dependent and give up on comparing to maps that are not equal-area. That still leaves a large permutation of more or less arbitrary choices.

Hopefully you can see that I am deeply dissatisfied with this state of affairs. Compared to the conformal situation, this is awful.

Is there hope for something better?

Map projection guru John Snyder theorized an equal-area analog to Chebyshev’s criterion in his 1988 paper, New equal-area map projections for noncircular regions, writing, “It seems that a region bounded by an isocol on an equal-area projection would be mapped with the minimum overall distortion of any equal-area projection of that region, as Chebyshev theorized and Gravé proved for conformal projections during the 19th century. I am not aware if proof exists of the equal-area counterpart, if in fact this is the case, but numerical checks I have made seem to support it.”

A proof does not exist. A refutation, however, does exist. Consider the Lambert equal-area projection, with isocols:

Lambert azimuthal equal-area projection

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The isocols on this projection are concentric circles, suggesting, by the Snyder conjecture, that the projection is optimal for any region bounded by one of the circles. Next consider the Wiechel projection, with isocols:

Wiechel equal-area projection

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The isocols on this projection are concentric circles, suggesting, by the Snyder conjecture, that the projection is optimal for any region bounded by one of the circles.

Wait. What? Yes. That’s right. Two very different projections—one with much greater distortion than the other—both have isocols of the same shape bounding the same regions. They cannot both be optimal. Therefore the Snyder conjecture is either false, or incomplete.

The problem with Snyder’s conjecture is that the boundary says nothing about the interior of an equal-area map. Contrast that to conformal maps, where the boundary says everything about the interior. Because the boundary says nothing about the interior of an equal-area map (and in fact, any arbitrary path in the projection says nothing about its interior), a given isocol does not describe a unique map. It describes practically nothing but itself.

What about other conjectures for a theory of optimal equal-area projections? Lev M. Bugayevski states in the 1995 book with Snyder Map projections—A reference manual, “Solving the problem of creating the best equal-area projections is in its initial stages. There are no works at present which have even a general solution of this problem...” Bugayevski further speculates that the optimal equal-area projections are those which are “most conformal”. This same sentiment was expressed to me in a discussion with the recently deceased Waldo Tobler, another prolific map projection researcher.

While it sounds good and plausible, how does one measure “most conformal”? I could imagine the same kinds of complexities and arbitrariness as we see up above in measuring angular distortion, especially across regions. Furthermore, there may be reasons to suppose that this notion is also false. Let me illustrate. As I noted in my first posting in this thread, the optimal conformal map for a world map permitting an interruption along one meridian is the Eisenlohr:

Optimal conformal map of the world

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To review, optimal in this context means, “The variation in flation is at a minimum in this map”. In the case of Eisenlohr, the outer boundary has a constant flation of 17+12√2, and this means that that is the maximum anywhere on the map. Everything inside is less. Now look at this map:

August epicycloidal

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This looks similar to Eisenlohr, but notice that it is smaller even though its minimum flation of 1.0 is the same in the center as Eisenlohr’s. Think carefully about that. We claimed that Eisenlohr was the least distorted conformal, but here we have another conformal map, August epicycloidal, that has less area inflation, overall, than Eisenlohr. In other words, August is “more equal-area” than Eisenlohr! How can this be?

The thing is, while August may inflate most of the world less than Eisenlohr, there is a small region at the poles where August’s flation goes considerably higher than Eisenlohr’s—nearly double the maximum of Eisenlohr’s. It does a worse job of keeping distortion under control. Now, it’s not wrong to claim that August is actually the “better” projection, and of course Herr August himself argued this vociferously with Herr Eisenlohr. Still, there is no theory for what makes a conformal map “most equal-area” and therefore “best”. All you can do is observe that one is more or less toward equal-area than another, and you will never know that you’ve reached optimal. For Chebyshev’s criteria, on the other hand, you know exactly when you’ve got what you seek.

I brought up this to illustrate that “most equal-area” does not apply to deciding which conformal map has the lowest distortion, and therefore perhaps we should not expect that the “most conformal” equal-area map is “the best”, either. What, then, are the true criteria?

I have my ideas, but nobody knows. I can give you a hint about what the optimal equal-area map of the whole world, split along one meridian, might look like, though. The next map is on what I call the “Dietrich–Kitada” projection. Bruno Dietrich was the author of a 1927 German text, Grundzüge der allgemeinen Wirtschaftsgeographie. The curious thing about this text is that it is illustrated with many full-page thematic maps, all in a single map projection, one whose construction method is… unknown, as far as I know. In late 1957, in a paper titled 世界全図に適する新図法の提出まで（続） (“Presenting new map projections for whole-world maps (continued)”), a Japanese mathematical cartographer named Kōzō Kitada (北田 宏蔵) described the projection he found in this German text, ascertained that it was equal-area, or at least nearly so, and set about formulating a recreation of it. He set the meridians to circular arcs, bounded the front hemisphere by a full circle, and determined the parallel shapes required to preserve area with these constraints. This is what he came up with:

Dietrich–Kitada

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The true optimal map will look somewhat different than this, but this is the gist.

And that’s what I have to say about optimal equal-area maps.

— daan

[Atarimaster]

Thanks a lot for the informative postings!
Will you at some point (here or elsewhere) elaborate on your ideas on the optimal equal-area map?

[Piotr]

So by this logic, circle–cropped Stereographic would be optimal for a conformal hemispheric map (and other small circles), and, generalizing the former, Lambert Conformal Conic would be optimal for conformal latitude cropped map, right?

[RogerOwens]

As I was saying before, raw information about max angular error, or min/max or max/min scale error, doesn't always best describe aesthetics, or even usefulness.

(That's why I suggested applying a rating-function to those raw measures.)

For example Tobler's Square CEA has more than half of the average min/max scale ratio at the map's various points, in comparison to that of Smythe-Craster.

But does anyone think that Tobler's Square CEA is more than half as good as Smythe-Craster, in terms of appearance, or usefulness for intuitive judgements regarding places' locations and relative locations?

But it gets worse than that:

On Tobler CEA, at the equator, the ratio of scales is equal to pi.

Sinusoidal, at lat 60 and lon 120, has point with a scale-ratio of about 8.

But which looks worse--Tobler CEA or Sinusoidal (or those named points on them)?

Michael Ossipoff

[daan]

As I was saying before, raw information about max angular error, or min/max or max/min scale error, doesn't always best describe aesthetics, or even usefulness.

As I was saying before, I don’t care about claims of æsthetics and “usefulness”. Those are personal, and if they’re not personal, then they’re statistical, but none of us have those statistics and so none of us are qualified to make statements about that on behalf of anyone else, let alone on behalf of science. Nor are we qualified to debate them.

Once we have such statistics and are motivated to act on them, we still need to know how to optimize for them.

— daan

[RogerOwens]

As I was saying before, raw information about max angular error, or min/max or max/min scale error, doesn't always best describe aesthetics, or even usefulness.

As I was saying before, I don’t care about claims of æsthetics and “usefulness”. Those are personal, and if they’re not personal, then they’re statistical, but none of us have those statistics and so none of us are qualified to make statements about that on behalf of anyone else, let alone on behalf of science. Nor are we qualified to debate them.

Once we have such statistics and are motivated to act on them, we still need to know how to optimize for them.

— daan

Sure, but would anyone, including you, disagree with the appearance-comparison (and probably the ease or difficulty of intuitively judging where to expect something to be), between Tobler CEA and Smythe-Craster that I mentioned? If not, then isn't it undeniable that the raw numbers would benefit from adjustment by a rating-function?

Of course it wouldn't be perfect, and there wouldn't be universal agreement about the exact coefficients for the approximate linear rating-function. But wouldn't it still be a clear improvement?

But, as for Tobler CEA vs Sinusoidal, I admit that I don't know how to fix that contradictory situation, in which a region with less min/max scale (on the Sinusoidal) looks better than a region with more min/max scale (on Tobler CEA). That puts the meaningfulness of the comparison of angular error, between different kinds of projections, in question. ...or so it seems to me.

Michael Ossipoff

[daan]

So by this logic, circle–cropped Stereographic would be optimal for a conformal hemispheric map (and other small circles), and, generalizing the former, Lambert Conformal Conic would be optimal for conformal latitude cropped map, right?

That’s true for the conformal hemispheric map. The conic case is complicated; there is no bounding isocol and so you cannot claim it is optimal for anything.

— daan

[RogerOwens]

daan--

It occurs to me that only for the rating of aesthetic appearance and realism, and only for scale-disproportion not caused by meridian convergence, does rating(min/max scale) go to to zero by the time min/max scale reaches the value it has at Peters' equator (That's .5).

Scale-disproportion caused by meridian-convergence, such as that of Sinusoidal doesn't make the map look ridiculously unrealistic like Peters, because it looks natural, because the side-view of a globe has it. So, CEA's plunge to zero appearance-ratng doesn't happen with scale-disproportion caused by meridian-convergence.

That seems to be the best way to explain and resolve the seeming contradictory results with Sinusoidal and Tobler CEA.

For intuitive judgement of positions, it's like you said: It doesn't matter whether or not the meridians converge, or by how much. What matters is just the flatness of the shape of the Tissot ellipse. That's what makes it more difficult to accurately judge where some point is, in a region that's distorted in that way. And, for that purpose, it doesn't matter whether the projection is Sinusoidal or CEA.

The min/max scale at Tobler CEA's equator is down to 1/pi. But it seems to me that the difficulty of accurately judging, intuitively without measurement, where some point is, in relation to a region, or another point, or lat/lon, in a region where min/max scale = 1/pi, is likely more than pi times more difficult than it would be where min/max scale = 1.

So I'd apply, to min/max scale, a ratings-function that reflects that.

Michael Ossipoff