Eisenlohr’s optimal conformal map of the world

[daan]

1. averaging over all azimuths, as Goldberg and Gott do with their closely related skewness;
2. selecting the value perpendicular to the isoline; or
3. selecting the maximum value at that point.

Are 2. and 3. always the same?

Yes, as a consequence of the directional derivative having value 0 perpendicular to the gradient, at which it is maximal in the uphill direction. This is not dependent on the mapping functions involved being conformal; only that their isocols are continuously differentiable locally.

— daan

[PeteD]

I’m skeptical of measures that depend on arbitrary decisions, such as the frequent “Let’s ignore the upper 15° [or 10° or whatever arbitrary value] of latitudes even in normal projections for computing distortion.

I agree, and the fact that pole lines must be excluded from the calculation is a major failing of the Airy criterion, along with being scale-dependent and treating inflation and deflation by the same factor differently.

While it’s true that most useful map projections do not have bizarre behavior beyond interruptions and infinite scales, the proposition of normalizing area measure requires more special pleading: It requires the projection not to have points with both infinite and infinitesimal area measure.

Could you give some examples of projections that have points with both infinite and infinitesimal area measure, please?

Using the gradient of flation, on the other hand, eliminates arbitrary choices. It also permits natural choices for “the point(s) of no distortion”, which on most conformal maps, coincides with the flation gradient being zero. This yields a natural analog to the unambiguous distortionless points on equal-area maps.

Yes, it all sounds very promising. My only reservation is that, like flexion and skewness, it must be more computationally expensive since the derivative must in general be calculated for every azimuth at every point.

[daan]

Could you give some examples of projections that have points with both infinite and infinitesimal area measure, please?

There are none in normal cartographic use.* The Littrow projection qualifies; I don’t know of any others, although Snyder’s GS50 is nearly such. In the case of Littrow, the areal measure is given by
s = (1 – sin²𝜆 ∙ cos²𝜑)/cos⁴𝜑
whereby area is infinite at
𝜑 = ±90°,
and zero when
𝜆 = arcsin(sec 𝜑), with real solution at 𝜑 = 0, 𝜆 = ±90°.

*Assuming that all equal-area projections with points or paths of infinite distortion never limit to both 0 and infinity in the same map at those points or paths. I haven’t checked for deviations from that.

— daan

[daan]

Using the gradient of flation, on the other hand, eliminates arbitrary choices. It also permits natural choices for “the point(s) of no distortion”, which on most conformal maps, coincides with the flation gradient being zero. This yields a natural analog to the unambiguous distortionless points on equal-area maps.

Yes, it all sounds very promising. My only reservation is that, like flexion and skewness, it must be more computationally expensive since the derivative must in general be calculated for every azimuth at every point.

I think it’s not so dire. You only need to compute the derivatives in two perpendicular directions; the remainder can be calculated as points on the ellipse circumscribing the two points and their reflections. I believe this follows from directional derivative theory, but I’m at the limits of my knowledge here.

— daan

[Milo]

Linear scale l is given by l² = a² cos² θ + b² sin² θ, where θ is the azimuth.

You're mixing up parametric angle with true (central) angle.

The Tissot ellipse (rotated to the x axis) is given by (x / a)² + (y / b)² = 1.

Linear scale at azimuth θ (relative to the direction of maximum scale) would then be the value l for which (l cos θ / a)² + (l sin θ / a)² = 1.

Solving this, I get 1 / l² = (cos θ / a)² + (sin θ / b)².

Flation p is given by p = a b.

Agreed.

Local areal distortion ε_p is given by ε_p = ln p = ln a b.

Although I myself am guilty of calling this "areal distortion" above, this is possibly not usage we want to encourage. More properly it's a useful intermediate value between flation and certain finalized distortion measures.

ε'_p = ln p' = ln a b c² = ln a b + ln c² = ε_p + ln c².

Thus, the rescaling factor is changed to a constant term. To obtain a metric that is scale-invariant, we need to apply a transformation which reduces constant terms to zero (which differentiation accomplishes, as do the other approaches you suggest).

1. Standard deviation of ε_p

2. Mean absolute deviation from the median of ε_p

Yes, these also work. Interesting.

3. Root mean square of ε_p

4. Absolute mean of ε_p

I do not see the value of these approaches.

And joke's on me, because they're actually the same as the first two

I do think that definitions 1/2 are more valuable in terms of highlighting the important properties of these metrics. Definitions 3 might be convenient for actual calculations (since it's a neat trick for computing the standard deviation of a dataset in a single pass, without needing to actually remember more than one data point at a time), but its scale-invariance is a non-obvious result.

5. Derivative of ε_p

As has been mentioned, the derivative of ε_p is another scale-invariant measure of how ε_p varies, and it is equivalent to (derivative of p) / p. The latter definition may appeal to those who find logarithms an unnatural function to apply to flation.

I don't see what's unnatural about it. People think about things like size according to logarithmic scales all the time.

But since, in this case, scale-invariance is intuitively obvious from either defintion, it matters little.

As you said, its value depends on the azimuth. Before we can start thinking about how to obtain a global value, we therefore first have to obtain a scalar local value. There are three obvious ways of doing this:

1. averaging over all azimuths, as Goldberg and Gott do with their closely related skewness;
2. selecting the value perpendicular to the isoline; or
3. selecting the maximum value at that point.

Are 2. and 3. always the same?

Provided that ε_p is continuously-differentiable, yes.

Wikipedia mentions this as an unnamed theorem.

I didn't remember this either, Daan had to point it out to me last time.

Once you have your scalar local value, you then have to decide whether to take the root mean square or the absolute mean or some other average in order to obtain a global value.

Right, that's an open question. I'm not going to worry about deciding what works best until we've worked out the local values for some more projections than just the Mercator.

At least, simply taking the maximum is unlikely to give good results, since that's infinite for both the Mercator and Lagrange projections. (Probably not Eisenlohr, but I'd have to check...)

I'm not interested in metrics that are only applicable to conformal projections.

My line of thinking is to try to come up with a metric that works well for conformal projections, try to come up with a metric that works well for equal-area projections, and then try to come up with a generalized metric that simplifies to either of the preceding two when restricted to projections in that category, but is also applicable to other projections. Work out the simple case before worrying about the more complicated one.

My previous attempt in this field was resolution-efficiency, which simplifies to maximum angle distortion in the equal-area case (minimized by circular Hammer), and simplifies to total normalized area in the conformal case (minimized by Lagrange), and which when applied to compromise projections, formally demonstrates the value of the plate carree projection (as well my projection over here, which I still wish more people would use...).

However, while I still maintain that resolution-efficiency is an important metric for certain applications, particularly related to data modelling or just source pictures for producing other projections, it doesn't actually produce the best-looking maps, which is particularly obvious in the equal-area case (circular Hammer looks bad).

Any decent metric for compromise projections would need to account for both area and angle distortion (and have a way of balancing their relative importance), not just measure one and ignore the other. When restricted to either equal-area or conformal projections, you can safely ignore the type of distortion that's just zero anyway.

Although I suppose you could try to investigate the Pareto front based on area and angle distortions... Indeed, I would argue that only projections on the Pareto front qualify as compromise projections.

Yes, it all sounds very promising. My only reservation is that, like flexion and skewness, it must be more computationally expensive since the derivative must in general be calculated for every azimuth at every point.

Ideally, you'd want to work out the derivative symbolically when you can, and then just use that formula. Numerical computation is a fallback for difficult projections, but something like Lagrange should be doable.

The Littrow projection qualifies;

Yeah, the Littrow projection is the one that came to mind for me as well.

And it's definitely a highly specialized projection that I wouldn't recommend for practical use unless you actually need the retroazimuthal property for some reason, but it's theoretically interesting (same goes for other retroazimuthal projections, really), in part because of its bizarre rarely-seen properties.

Outside of the Littrow projection, the vast majority of conformal and compromise projections in use do have nonzero minimum flation. The orthographic projection has zero minimum flation, but finite maximum flation.

I don’t know of any others, although Snyder’s GS50 is nearly such.

Over the area it's optimized for, or when pathologically extended to the whole globe?

*Assuming that all equal-area projections with points or paths of infinite distortion never limit to both 0 and infinity in the same map at those points or paths. I haven’t checked for deviations from that.

Huh?

Equal-area projections, per definition, have the same area measure everywhere, and it is therefore never infinite or infinitesimal. Even at the singularities, where linear scale can reach infinite values, i.e. the Tissot ellipse's axes are infinite a and infinitesimal b, the product a*b is still the same finite value. Though by the same token, any equal-area projection that reaches infinite angle distortion somewhere (even just at a single point) will have both infinite and infinitesimal linear scales.

I think it’s not so dire. You only need to compute the derivatives in two perpendicular directions; the remainder can be calculated as points on the ellipse circumscribing the two points and their reflections. I believe this follows from directional derivative theory, but I’m at the limits of my knowledge here.

Wikipedia helps out here too: "If f is differentiable, then the dot product (∇f)_x ⋅ v of the gradient at a point x with a vector v gives the directional derivative of f at x in the direction v."

This has interesting corrolaries with regard to the skewness metric that PeteD attributes to Goldberg and Gott. That metric would be the average of the above value as v varies over all unit vectors. Since inner products and means are both linear, we find (using PeteD's angle bracket notation):
⟨∇f(x) ⋅ (cos θ, sin θ)⟩ = ⟨∇f(x₁ ⋅ cos θ + ∇f(x₂ ⋅ sin θ)⟩ = ∇f(x₁ ⋅ ⟨cos θ⟩ + ∇f(x₂ ⋅ ⟨sin θ⟩ = ∇f(x) ⋅ ⟨(cos θ, sin θ)⟩
This causes a problem, since the average of the latter vector is simply zero, so something is missing about the definition PeteD gave. Presumably, we're supposed to take the mean of the absolute value of the directional derivatives, rather than just of the directional derivatives themselves. (Or take the mean only over the half of the directions where the derivative is positive.) This slightly complicates things, but ultimately not that much, and the same sort of logic can eventually be used to show that:
⟨|∇f(x) ⋅ (cos θ, sin θ)|⟩ = |∇f(x)| ⋅ 2/π
So this metric is simply the same as the "perpendicular to the isoline" one, rescaled by a constant factor.

[PeteD]

I think it’s not so dire. You only need to compute the derivatives in two perpendicular directions; the remainder can be calculated as points on the ellipse circumscribing the two points and their reflections. I believe this follows from directional derivative theory, but I’m at the limits of my knowledge here.

Right.

You may or may not remember that I've been playing with Goldberg and Gott's IDL code. A while ago, I modified it mainly by commenting out the parts that calculate flexion and skewness because I'm not convinced that low flexion and skewness are particularly desirable (except low flexion for applications that require the gnomonic projection and low skewness to the extent that it's somewhat correlated with low values of the Airy-Kavrayskiy criterion, but then it's quicker to just calculate the Airy-Kavrayskiy criterion instead).

These modifications achieved a sevenfold increase in speed, and this is what I was thinking of. However, it seems the extra computation time for flexion and skewness was due to calculating two extra derivatives at each point rather than an extra derivative for each azimuth at each point. I also made some other modifications to the code at the same time, in particular rewriting loops as matrix operations, which may have contributed more to the sevenfold increase in speed than I'd realized.

[PeteD]

You're mixing up parametric angle with true (central) angle.

Oops! Thanks for pointing that out. I've just deleted this line from my previous post since I never used l after defining it anyway.

Although I myself am guilty of calling this "areal distortion" above, this is possibly not usage we want to encourage. More properly it's a useful intermediate value between flation and certain finalized distortion measures.

Whether "(local) areal distortion" refers to ln p, ln² p, abs(ln p), p - 1, (p - 1)², abs(p - 1), max(p, 1/p) - 1 or abs(p - 1) / (p + 1) is largely a question of semantics since they are all measures of distortion of areas.

And joke's on me, because they're actually the same as the first two

For me, it's not at all intuitive without doing the maths that these should lead to the same results, but maybe it is intuitive for someone with a deeper insight.

I do think that definitions 1/2 are more valuable in terms of highlighting the important properties of these metrics.

I agree. I like the fact that there's no explicit rescaling.

Definitions 3 might be convenient for actual calculations (since it's a neat trick for computing the standard deviation of a dataset in a single pass, without needing to actually remember more than one data point at a time), but its scale-invariance is a non-obvious result.

Definitions 3 are 4 are how these metrics have historically been derived, and I think that this hasn't helped their cause since the rescaling is in a way arbitrary, although I would maintain that rescaling in order to minimize the global distortion value is the least arbitrary rescaling.

Definitions 3 and 4 are also useful to highlight the link with global angular distortion E_a defined as

E_a² = ⟨ε_a²⟩ or E_a = ⟨abs(ε_a)⟩, where ε_a = ln a/b.

I can't think of a definition of angular distortion that's analogous to the derivative of flation in the same way that E_p and E_a are analogous to each other. I don't see any obvious value in taking the derivative of a/b or ln a/b. This is perhaps my biggest outstanding reservation about using the derivative of flation.

I don't see what's unnatural about it. People think about things like size according to logarithmic scales all the time.

"Unnatural" probably wasn't the best word. What I meant was that people who are less mathematically inclined might be averse to metrics involving functions that they don't understand or that they wouldn't think of using themselves. A few years ago, Tobias said that he liked Capek's Q because he could actually understand it.

This causes a problem, since the average of the latter vector is simply zero, so something is missing about the definition PeteD gave. Presumably, we're supposed to take the mean of the absolute value of the directional derivatives, rather than just of the directional derivatives themselves.

Yes. Sorry for the omission.

So this metric is simply the same as the "perpendicular to the isoline" one, rescaled by a constant factor.

I must have skipped over this part of their paper when I first read it because I didn't realize this, but there it is on page 11 for all to see.

[Milo]

You may or may not remember that I've been playing with Goldberg and Gott's IDL code.

I remember no such thing, but searching the forum, it turns out, I did in fact post in that thread back in the day.

What does "IDL" stand for?

A while ago, I modified it mainly by commenting out the parts that calculate flexion and skewness because I'm not convinced that low flexion and skewness are particularly desirable (except low flexion for applications that require the gnomonic projection and low skewness to the extent that it's somewhat correlated with low values of the Airy-Kavrayskiy criterion, but then it's quicker to just calculate the Airy-Kavrayskiy criterion instead).

I don't know exactly what this "flexion" is, but by the context, I'm guessing it's a measure of curvature?

I agree that this does not seem to be an especially useful property for a map as a whole. Some maps, such as tangent conics (conic projections with both standard parallels equal), are notable for being curvature-preserving at the standard parallel, but not over the whole thing.

Since curvature is essentially a continuous/infinitesimal form of angular deformation, I think that minimizing curvature error is rather pointless anywhere that the map is not already conformal. (Which, of course, conics are at their standard parallel.)

On another note, I must say that "skewness" is a rather weird name for what's essentially a measure of area distortion, if it indeed is the same thing I've reinvented here as you claim.

Definitions 3 and 4 are also useful to highlight the link with global angular distortion E_a defined as

E_a² = ⟨ε_a²⟩ or E_a = ⟨abs(ε_a)⟩, where ε_a = ln a/b.

I can't think of a definition of angular distortion that's analogous to the derivative of flation in the same way that E_p and E_a are analogous to each other. I don't see any obvious value in taking the derivative of a/b or ln a/b. This is perhaps my biggest outstanding reservation about using the derivative of flation.

Indeed, angular distortion is better measured as a/b or a monotonic function thereof (for example, the 2*atan(sqrt(a/b)) - pi/2 = asin(1-2/(a/b+1)) = asin((a-b)/(a+b)) metric that everyone seems to like to use despite its computational complexity). The existance of such monotonic functions, however, does make it somewhat arbitrary exactly which value you're taking the average of (or which average you're taking). I don't think that adding the angular distortions at different points, or their squares, or whatever, produces any naturally-meaningful value.

The advantage about using the derivative of logarithmic flation is that it's a local measure, which can show how badly-distorted the map is in any small area (and which, as Daan points out, can be used to determine the "point of zero distortion"). By contrast, standard deviation of logarithmic flation is a global value that can only meaningfully be computed for the map as a whole, meaning that taking the same projection but extended to a larger or smaller region would produce completely different results.

Note that conformal projections already have the property of any small local region on the projection looking decent, even if at different scales (which is one reason why people like to use conformal projections, even if I'd argue that with modern computing power most high-quality local maps wouldn't actually be made by cutting them from a larger world map), therefore the derivative-of-logarithmic-flation measure is useful because it essentially expands on this observation, telling you "how small" a region has to be for this property to hold. In the Mercator projection, Africa looks pretty good, but Greenland has somewhat noticeable distortion just between the north and south ends of the island, despite being much smaller than Africa (well, smaller than Africa on the real sphere...). This is because the area distortion measure under discussion is greater over Greenland than over Africa.

[PeteD]

What does "IDL" stand for?

Interactive Data Language.

I don't know exactly what this "flexion" is, but by the context, I'm guessing it's a measure of curvature?

Yes, curvature of geodesics. Goldberg and Gott present it as a counterpart to skewness in the way that areal and angular distortion are counterparts to each other. It's zero everywhere for the gnomonic projection, which isn't a projection that you'd want to emulate apart from for some very specific applications. I like to think of flexion as "gnomonicness" to highlight its lack of desirability in most cases.

On another note, I must say that "skewness" is a rather weird name for what's essentially a measure of area distortion, if it indeed is the same thing I've reinvented here as you claim.

(derivative of flation) / flation is a measure of areal distortion. Skewness is (derivative of linear scale) / linear scale, which is only a measure of areal distortion in conformal projections. For equal-area projections, it's non-zero, so it can't be a measure of areal distortion in general. In fact, it tends to be lowest for compromise projections.

This has got me thinking: instead of trying to come up with a good measure for areal distortion and an analogous measure for angular distortion and then combining them to use as an overall distortion metric, maybe it's better to come up with a single measure that serves as a measure of areal distortion in conformal projections, a measure of angular distortion in equal-area projections and a measure of both in compromise projections. Could skewness be that measure? It certainly serves as a measure of areal distortion in conformal projections. I'll have to think about the rest.

I don't think that adding the angular distortions at different points, or their squares, or whatever, produces any naturally-meaningful value.

This whole discussion arose from the question "Is the Mercator or the Lagrange projection better for a conformal world map?". Using local distortion measures, we can plot distortion charts like the one that you made, which are very informative but ultimately can't answer this question because although they show that the Lagrange projection has higher distortion over most of the globe, in some of the areas where the Mercator projection has higher distortion, it has far higher distortion. This question can ultimately only be answered using a global distortion metric. While it's true that we may never agree on one, that's no reason not to try.

taking the same projection but extended to a larger or smaller region would produce completely different results.

Isn't this desirable? If you're asking the question "What's the best projection for this map?", then the answer will depend very strongly on the region that you're mapping, and an overall distortion metric should reflect this.

[PeteD]

As has been mentioned, the derivative of ε_p is another scale-invariant measure of how ε_p varies, and it is equivalent to (derivative of p) / p ... As you said, its value depends on the azimuth. Before we can start thinking about how to obtain a global value, we therefore first have to obtain a scalar local value. There are three obvious ways of doing this:

1. averaging over all azimuths, as Goldberg and Gott do with their closely related skewness;
2. selecting the value perpendicular to the isoline; or
3. selecting the maximum value at that point.

So 2. and 3. are always the same, while 1. differs only by a constant factor of 2/π, so it doesn't matter which option we take.

Once you have your scalar local value, you then have to decide whether to take the root mean square or the absolute mean or some other average in order to obtain a global value. Goldberg and Gott use the absolute mean for flexion and skewness despite using the root mean square for areal and angular distortion.

The integral of the square of these metrics doesn't converge for the equirectangular or Mercator projections, and I dare say for many others, so the root mean square is out. The number of arbitrary choices is dropping like flies.