A new map metric: resolution-efficiency

[Milo]

The resolution at a point is defined as the semiminor axis of its Tissot indicatrix, as measured when the map is scaled so its total area is equal to the original area of the mapped region. (Or: the semiminor axis of its Tissot indicatrix, divided by the square root of the ratio of the projected area to the original area.)

The resolution-efficiency of a projection is defined as the lowest resolution of any point.

Resolution-efficiency can take values in [0,1]. A value of exactly 0 is reached by many common projections, namely those which require infinite area (stereographic, Mercator, gnomonic) and those which have a semiminor axis of zero somewhere (cylindrical and azimuthal equal-area, orthographic, Mollweide and all other pseudocylindrical equal-area projections that don't have an angle at the poles). A value of exactly 1 cannot be reached by a Euclidean projection (since it represents no distortion), but it is possible to come arbitrarily close by having lots of interruptions, or by projecting only a small region rather than the whole sphere.

Intuitively, having a high resolution-efficiency guarantees that the projection will not destroy important details, even though it may exaggerate unimportant ones. Thus, it's a measure of how much detail you can reliably show on a map of a given size, or of how small you can squeeze a map (to avoid wasting space, either physical space on a page or digital space used to store an image file) while still retaining a certain level of detail. One example where this property is useful is for storing "raw input data" that might subsequently be converted into a different projection (for example, consider how the cylindrical equal-area projection squashes the poles, and this lost detail cannot be recovered if you then want to reproject as a polar-aspect azimuthal projection). However, it's also useful for end-user display maps, where it ensures that any features the viewer might be interested in remain recognizable, even if they are distorted.

When applied to equal-area projections, maximizing resolution-efficiency is equivalent to minimaxing angle distortion. If done on the meridian-interrupted sphere, this gives the circular Hammer projection. This does seem to be unfortunate as it doesn't intuitively look like a very good projection... but no-one has yet devised a better optimality criterion for equal-area projections, and this one should at least work better for regions that are closer to circular, such as regular polygons. (After all, the azimuthal equal-area projection is optimal for circles.)

When applied to conformal projections, maximizing resolution-efficiency is equivalent to minimizing mean normalized area distortion (in fact, it's pretty much a generalization of the idea that isn't restricted to conformal projections). If done on the meridian-interrupted sphere, this gives the Lagrange projection (discussed here, here and here). This is different from minimaxing area distortion (as the Eisenlohr projection does).

However, it is also possible to optimize purely for resolution-efficiency on its own, without restricting to equal-area or conformal projections. Indeed, my original purpose in devising this criterion was to come up with a good optimality criterion that can be applied to compromise projections, and more particularly to explain what's so good about equidistant projections, without making reference to which specific lines they are equidistant along (since they are in fact not equidistant between most points, greatly limiting their usefulness for actual distance-measuring).

The plate carree projection (specifically, and not cylindrical equidistant projections with standard parallel away from the equator) obviously has optimal resolution-efficiency among cylindrical projections, and I suspect that it is optimal among all full-world projections with a convex shape, but I haven't proven this (there can, at least, exist other projections which are tied with it). The azimuthal equidistant projection is optimal among azimuthal projections, but is not as good as the plate carree projection (except when interrupted into two-hemisphere form). In any case, for the applications where resolution-efficiency is important, having a rectangular shape is often also useful in its own right.

An interesting property of this metric is that equal-area and conformal projections that are in some sense "counterparts" to each other (such as azimuthal equal-area and stereographic hemispheres, or the circular Hammer and Lagrange projections mentioned above) often have the same resolution-efficiency, suggesting that these are somehow "equally far apart" from the ideal compromise projection in the middle. The orthographic and gnomonic projections (when restricted to a circle smaller than a hemisphere, so that they have nonzero resolution-efficiency) are also found to be "counterparts" in this sense, which feels appropriate.

A related concept is that of constant-resolution projections. A projection is defined as constant-resolution if its Tissot indicatrices have the same semiminor axis everywhere. The well-known resolution-efficiency-optimizing projections (plate carree, azimuthal equidistant) are also constant-resolution, and vice versa, but there is no obvious guarantee that this will always be the case.

Some examples of resolution-efficiency of common projections:

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Full-world projections:
Plate carree:          sqrt(2/pi)          ~= 0.7979 (general-case cylindrical equidistant: sqrt(2/pi*cos(standard))
Azimuthal equidistant: 2/pi                ~= 0.6366
Lagrange:              1/2                  = 0.5
Circular Hammer:       1/2                  = 0.5
Classic Hammer:        1/2/sqrt(2)         ~= 0.3536
Sinusoidal:            (sqrt(pi^2+4)-pi)/2 ~= 0.2913

For Eisenlohr, I've estimated approximately 0.3893 from pixel-counting a screenshot, but I don't have a closed form.

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2-hemisphere azimuthal:
Equidistant:   sqrt(8)/pi ~= 0.9003
Equal-area:    sqrt(1/2)  ~= 0.7071
Stereographic: sqrt(1/2)  ~= 0.7071

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Azimuthal arbitrary-radius:
Equidistant:   sin(r/2) / (r/2)
Equal-area:    cos(r/2)
Stereographic: cos(r/2)
Orthographic:  sin(r/2) / (tan(r)/2)
Gnomonic:      sin(r/2) / (tan(r)/2)

I've also computed some theoretical optima for polyhedral projections. However, to my knowledge, no-one has actually constructed these projections. Well, actually, I do have one projection whose resolution-efficiency matches the theoretical optimum. It's really bad (it dramatically fails the "smooth except at the vertices" criterion I asked for here, and it's also not constant-resolution, unlike the successful equidistant projections), but it serves as a proof of concept that the theoretical bounds I computed are in fact reachable (for the non-equal-area case).

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Optimal polyhedral, theoretical:
Dihedron:            sqrt(8/pi/n / tan(pi/n))
Triangular dihedron: sqrt(8/3/pi/sqrt(3))                  ~= 0.7001
Square dihedron:     sqrt(2/pi)                            ~= 0.7979 (same as plate carree)
Pentagonal dihedron: sqrt(1.6/pi * sqrt(1 + 2/sqrt(5)))    ~= 0.8372
Hexagonal dihedron:  2/sqrt(pi*sqrt(3))                    ~= 0.8574
Tetrahedron:         sqrt(pi/3/sqrt(3))/acos(1/sqrt(3))    ~= 0.8139
Diamond:             sqrt(pi/6/sqrt(3))/acos(sqrt(2/3))    ~= 0.8933
Cube:                sqrt(8/3/pi)                          ~= 0.9213
Icosahedron:         sqrt(pi/15/sqrt(3))/acos(phi/sqrt(3)) ~= 0.9531
Dodecahedron:                                              ~= 0.9699

The square dihedron (i.e., a hemisphere-in-a-square projection, a la Peirce/Guyou, Gringorten, etc.) having the same value as plate caree is interesting. When unfolded, they would both produce the same shape (a 2:1 rectangle) with the same resolution-efficiency, but with distortion spread out differently (in particular, it should be possible for a well-made hemisphere-in-a-square projection to have an upper bound on how much it distorts anywhere, by contrast with cylindrical projections which have infinite distortion at the poles).

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Optimal equal-area polyhedral, theoretical:
Dihedron:            sqrt(pi/n/tan(pi/n)/2)
Triangular dihedron: sqrt(pi/6/sqrt(3))                    ~= 0.5498
Square dihedron:     sqrt(pi/8)                            ~= 0.6267
Pentagonal dihedron: sqrt(pi/10/sqrt(5-2*sqrt(5)))         ~= 0.6576
Hexagonal dihedron:  sqrt(pi/4/sqrt(3))                    ~= 0.6734
Tetrahedron:         sqrt(sqrt(3)/pi)*acos(1/sqrt(3))      ~= 0.7093
Diamond:             sqrt(pi/8*sqrt(3))                    ~= 0.8247
Cube:                sqrt(6/pi)*acos(sqrt(2/3))            ~= 0.8506
Icosahedron:                                               ~= 0.9191
Dodecahedron:                                              ~= 0.9353

[dummy_index]

interesting...but it seems to depend on the simplicity of projection formula...

The plate carree projection (specifically, and not cylindrical equidistant projections with standard parallel away from the equator) obviously has optimal resolution-efficiency among cylindrical projections, and I suspect that it is optimal among all full-world projections with a convex shape, but I haven't proven this (there can, at least, exist other projections which are tied with it). The azimuthal equidistant projection is optimal among azimuthal projections, but is not as good as the plate carree projection (except when interrupted into two-hemisphere form). In any case, for the applications where resolution-efficiency is important, having a rectangular shape is often also useful in its own right.

found it!

EAzEdCapsule.png (91.42 KiB) Viewed 8877 times

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Full-world projections:
Capsule:               4/sqrt(4 pi + pi^2) ~= 0.8445
Plate carree:          sqrt(2/pi)          ~= 0.7979 (general-case cylindrical equidistant: sqrt(2/pi*cos(standard))
Azimuthal equidistant: 2/pi                ~= 0.6366

[Milo]

Ah, I see what you did there. Clever!

I worked it through and everything checks out.

Its derivative does have a discontinuity at the seams, but it's the semimajor axis that's affected by that, not the semiminor, so it's a non-issue by the definition I gave. And since my conjecture was wrong, there's nothing ruling out even better possibilities now.

I observe that a lot of popular projections have a roughly similar appearance to your capsule (flat or near-flat poles but rounded corners). Eckert III, Robinson, that other Snyder projection, Winkel Tripel... Maybe they're on to something.

Even Aitoff, if you stretch what counts as "near-flat poles". Actually, I should compute Aitoff's value. It's pseudo-equidistant, so it has pretty good odds of doing well. Aitoff has a total area of pi^3/2, but its equidistance isn't quite as good as it could be, with semiminor axes going as low as 0.5. That translates to a resolution-efficiency of sqrt(2)/pi ~= 0.4502. Nope. Better than Hammer, but still far worse than the plain equidistant projections or your capsule.

But the circular Hammer projection worked better than the 2:1 ellipse. What happens if we try that with Aitoff? Area is now pi^3, but from what I can tell, semiminor axis never dips below 1 now. (Makes sense, the longitude halving can at most halve any distances, and then we double it again.) That means 2/pi ~= 0.6366. That's... actually the same value as the azimuthal equidistant, even though it obviously isn't the same projection.

Let me try Eckert III. Area is the same as your capsule: pi^2+pi^3/4. Minimum semiminor axis... hard to compute exactly, but it seems to be somewhere around 0.86. Definitely less than 1, so it'll do worse than the capsule.

[PeteD]

Very interesting. If I'm not mistaken, your resolution efficiency seems to be a variant of Michael Ossipoff's min scale, presented for example at viewtopic.php?f=8&t=802 but also on numerous other threads of this forum, possibly not always in an entirely consistent manner.

Is resolution efficiency also used more widely? Or just by Michael Ossipoff and yourself?

Edit: fixed the broken link -- thanks, Tobias and Milo, for pointing that out!

[Milo]

Your link is broken. Here's a corrected version:

viewtopic.php?f=8&t=802

That post was a little hard to parse (it looks like someone crudely copy+pasted text from another source without retaining the original formatting or images?), but yes, it does look like it's doing something similar. The difference is that instead of normalizing maps by their own area, they're normalized by the area of the smallest square bounding box that fully contains them. Which seems inefficient when I put it that way - even in the applications where you'd care about bounding boxes, you'd at least want to allow rectangular ones. (Computer image formats, for example, are usually only capable of encoding a rectangular area, even if some of that area is transparent. Pages in a book are usually rectangular but rarely square.)

Though he actually defines several different concepts of "relevant reference-distance", and the last one mentions "maybe [...] so that its area is all that’s relevant to its fit there", which I guess might actually correspond to my metric after all, even though I don't think that describing it in terms of "mean-reference-distance" rather than area is a good idea.

He also arbitrarily constrains his definitions as "out to lat 71.2, the lat of North Cape", which serves to mitigate the extreme distortion of many map projections near the poles, but is clearly not suitable as a universal metric: some maps are specifically intended to show the polar regions (usually polar-aspect azimuthal), some maps don't get anywhere near the polar regions (regional map of Africa, say), some maps are of planets/moons other than Earth on which the particular geographical extremes of Earth would be irrelevant, and even on Earth, there are "northernmost points" more or less far north than those of Norway (depending on which continents and islands you're interested in), whereas there might also be parts of the map far closer to the equator that you're still not interested in (the middle of the Pacific Ocean for a land map ignoring minor islands, for example) yet which this metric would still count. My definition is equally applicable to both world and regional maps, and I have shown that some projections demonstrate pretty good resolution-efficiency even when applied to the whole world without such special pleading.

The other metric from that link that I've given some thought to, which he calls "min/max scale" (I'd probably have gone with "scale ratio" or "scale range" or something, to avoid confusion with minimax optimizations), would be described in my terms as the smallest semiminor axis of any point divided by the largest semimajor axis of any point (or maybe I'd use the reciprocal, for that "large numbers are scary" feel). For conformal projections, this is the metric whose optimization produces the Eisenlohr projection (as opposed to resolution-efficiency, which produces Lagrange). For equal-area projections, both metrics actually end up being the same thing (or rather, resolution-efficiency as I chose to define it here is equal to the square root of min/max scale, but that's the same thing as far as maximizing it goes).

In this post (scroll down to the second quote), I presented a parametric metric that (up to monotonic transformations) corresponds to resolution-efficiency when p = 1, and to min/max scale when p = infinity. However, it applies only to conformal projections. I have not yet found a way to make a similar generalization work for non-conformal projections in a way that doesn't feel forced/arbitrary. Also it still wouldn't serve a point for equal-area projections anyway, since as noted above, both metrics are the same there, and so presumably any intermediate ones would be as well. (Also note that while that post takes after dummy_index's terminology in calling what it's doing a "norm", I've since concluded that it's actually better understood as a power mean.)

Is resolution efficiency also used more widely? Or just by Michael Ossipoff and yourself?

Dunno.

I invented it myself. It's simple enough that it wouldn't be a huge surprise if it turns out someone else independently came up with the same idea, but I hadn't personally seen any previous discussion of it while studying map projections.

I've been sitting on it for a while before I decided to post it here, but this thread is my first time mentioning it to anyone other than myself.

[daan]

The resolution at a point is defined as…

This is a very interesting metric, especially given how the analogous equal-area and conformal projections yield the same measure. However:

[For Eisenlohr, I've estimated approximately 0.3893 from pixel-counting a screenshot, but I don't have a closed form.

As you noted here, the real value is
2/[(3 + 2 √2) √(4 ln 2 - 2)] ≈ 0.3903951

My attempt at producing an equal-area projection analogous to the Eisenlohr does not yield the same measure as Eisenlohr. Using the Eisenlohr’s resolution efficiency, the maximum angular deformation for an equal-area projection would be 94.69°. I don’t have an analytic result for my projection, but it’s something a little under 81°.

What does this mean? Probably nothing. Just noting it.

Cheers,
— daan

[quadibloc]

The capsule projection has inspired me to concieve of a projection that would aid the quest for resolution-efficiency.
The Aitoff projection is already the analogue of the Hammer projection which is based on the azimuthal equidistant instead of the Lambert azimuthal equal-area.
Clearly, to improve on the capsule projection, one should replace the plate caree segment by an interrupted version... of the analogue of the Adams projection that is based on the azimuthal equidistant instead of the Lambert azimuthal equal-area. That would create a projection somewhat similar to the Oxford projection which was designed by Guy Bomford.

[Milo]

This is a very interesting metric, especially given how the analogous equal-area and conformal projections yield the same measure. However:

My attempt at producing an equal-area projection analogous to the Eisenlohr does not yield the same measure as Eisenlohr.

For what it's worth, the equal resolution-efficiency for conformal and equal-area projections in some of the simplest cases (azimuthal, Hammer-like azimuthal modifications) is more of an interesting coincidence than something with theoretical justification. It stands to reason that certain map shapes may simply be better-suited for one or the other property, although it's then noteworthy that these "most basic" projection types are equally suitable for both.

In particular, note that my calculation of a theoretic optimum resolution-efficiency for an equal-area square dihedral projection, 0.6267, is lower than my calculation for the resolution-efficiency of the conformal square dihedral projection, 0.6760. Although I can't guarantee that this upper bound for the equal-area case is actually reachable (though I think it probably is), I can easily prove that it cannot be exceeded, and therefore no equal-area projection can reach the same value as the has-actually-been-constructed conformal projection for this shape.

Of course there is some abiguity in what projections are considered "analogous to" each other. There doesn't (currently!) exist any formal theory of "dual projections", so it's more a matter of looking similar in some aspect that we happen to care about.

The capsule projection has inspired me to concieve of a projection that would aid the quest for resolution-efficiency.

For what it's worth, I'm convinced that my initial search space of "all projections with a convex shape" was naive to begin with. I think you could get resolution-efficiency arbitrarily high by clever application of a long and thin "mummy wrap" projection shape, which is technically convex, but not a shape any sane person would use.

Instead, it would be more sound to search in the space of "projections of the meridian-interrupted sphere", which is clearly the category of projections that most people favor anyway.

(The capsule projection meets both criteria, so it serves as a counterexample either way!)

Clearly, to improve on the capsule projection, one should replace the plate caree segment by an interrupted version...

I've already established that interrupted projections can push resolution-efficiency arbitrarily high, as with sinusoidal globe gores. So unless you have some way to glue the interruptions back together later, the point is to see how high you can get without excess interruptions.

of the analogue of the Adams projection that is based on the azimuthal equidistant instead of the Lambert azimuthal equal-area.

Umm, what? The only Adams projections I know of are conformal, and therefore obviously not based on the Lambert azimuthal equal-area.

You can't mean Aitoff either, since that one's already based on the aximuthal equidistant, it's redundant to mention that twice.

My next-best guess on what you meant would be the Albers conic projection, which shares a couple of letters and is equal-area, but I don't see its relevance here, and, again, there already is a conic equidistant projection so you could have just named it directly.

That would create a projection somewhat similar to the Oxford projection which was designed by Guy Bomford.

Just to be clear(er), are you talking about this thing?

[mapnerd2022]

No, he's talking about another projection by Guy Bomford which is aproximately equal area. Quadibloc has devised a strictly equal area version of it. it's (further) interrupted and condensed. Oscar Sherman Adams has independently presented in 1945 a projection called the Quartic Authalic which had already been devised by Karl Siemon in 1937. The Quartic Authalic is pseudocylindrical and equal-area. It is derived from azimuthal principles. It has it's parallels spaced like in the equatorial aspect of the Lambert Azimuthal Equal-Area. Here it is: https://mapthematics.com/ProjectionsLis ... 20authalic

[Milo]

Thanks. Found it!

So that's the "Adams projection" he was talking about, though that link doesn't explain the construction.