A flawed "cartometry"

[Atarimaster]

Hello,

I’ve got a problem which is not related to Geocart, but since Geocart is an important part in the process and moreover, because I don’t know any other forum to ask (anymore), I’m posting it here.

It started I wanted to know Capek’s Q [1] for several projections.
Since Geocart doesn’t compute that value (hint, hint ), I thought of a way to obtain it by doing a bit amateurish cartometry. It works well on equal-area projections, but utterly fails for compromise projections, and I can’t see why.
So I’m going to tell you what I’m doing, and hopefully someone can provide me a clue at which point I’m doing a mistake.

Capek defined Q as the percentage ratio of the area represented in the map with permissible distortion to the area of the whole world.
Both limits can be set arbitrarily, but I used the same that Capek used in his paper:
Maximum angular distortion 40° and 1.5 multiple of the smallest areal distortion were used as the distortion limits.

Capek calculated Q for 100 projections by using cartometry and additionally for 30 of them, by computer processing derived from the projection’s formula.

Let me show you what I’m doing using Eckert IV as example. Being an equal-area projection, we just have to take the angular distortion into account.
To visualize maximum angular distortion 40°, I’m using a metric lines database containing the following line:

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<isocol max_deformation='40' />

The database is added to a map using Eckert IV projection showing only the map boundaries. No graticule, no other databases. The map is exported as TIFF image, looking like this:

eckert-4.step1.png (6.24 KiB) Viewed 2052 times

(Note: The images I used for my "cartometry" are actually larger than the ones I’m showing here, which I scaled down.)

In an image processor, the image is colorized: Areas with an angular distortion above 40° are filled red, the others green:

eckert-4.step2.png (8.79 KiB) Viewed 2052 times

Now, I’m using a script that counts all pixels that are not white (= the area of the whole world) and all pixels that are green (= the area represented in the map with permissible distortion), so I can calculate Q:

Code: Select all

(number of green pixels) / (number of non-white pixel) * 100

… which is the Eckert IV image that I used:

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599395/735300*100 = 81.517067863457

Since Capek works with one decimal, my result gets rounded to 81.5.
In Cape’s paper, Eckert IV is listed with 81.9 (Q set by computer) and 82.5 (Q set by cartometry). So my result deviates by 0.4/1.0, which can easily explained by the fact that my approach certainly isn’t accurate. However, it’s close enough for my purposes.

I repeated this for various other equal-area projections (e.g. sinusoidal, Putnins P2, Mollweide, Hufnagel 10 & 11, Wagner IV & VII…) with similar deviations.
My approach works! Yay me!

BUT…
Then I moved on to the compromise projections.
Now I needed to visualize the 1.5 multiple of the smallest areal distortion.
In Wagner VI, the smallest areal distortion is 1.0 (= equivalent, at the center of the map).
So the line in the metric lines database is:

Code: Select all

<isocol area='1.5' />

So now, the color green is applied to the areas that are both below an angular distortion of 40° AND an areal distortion of 1.5.
This is how the maps looks like before and after the colorization:

wagner-6.both.png (18.14 KiB) Viewed 2052 times

I’m running my script and GULP!
Capek lists 80.4 (computer) and 79.5 (cartometry), but my result is 66!
Did my script do something wrong in counting the non-white and green pixels?
I confirmed the results using this technique. No, my script was doing just fine. And if this hadn’t been the case, the results for the equal-area projections would have been wrong, too. To the script, it doesn’t matter a bit whether the projection is equal-area or not, it just count pixels of various colors.

And of course, I tried this with other compromise projections. All of them were deviating terribly from Capek’s results.
So, what am I doing wrong??


P.S.: Due to my cartometry, which as I’ve said seems to be quite accurate for equal-area projections, the Q of van Leeuwen projection is 91.7 and thus, higher than for any projection that was examined by Capek.
P.P.S.: Among the interrupted projections Capek lists a projection called BSAM. Yet the only projection bearing this name that I know of is an uninterrupted cylindric projection. So which projection is he referring to?



[1] Capek, Richard 2001:
Which is the Best Projection for the World Map?
https://icaci.org/files/documents/ICC_p ... f24014.pdf

[Atarimaster]

Aaah, I think now I see my fallacy.
Q is the percentage ratio of the area represented in the map with permissible distortion to the area of the whole world.
But my cartometry is calculating the percentage ratio of the area represented in the map with permissible distortion to the area of the whole map.
This is of course something different and inevitably leads to a lower Q value on compromise projections, and also explains why the values for equal-area projection were right.
Damn.

Hmmm. I think I still could get the right Q if I use Geocart’s raster reprojection engine to reproject e.g. the red and green Wagner VI image to an equal-area projection and run the pixel-counting script on the result, right?

That’s more work than I hoped.
Nonetheless, I’m going to try. But not today.

Kind regards,
Tobias

[daan]

Hmmm. I think I still could get the right Q if I use Geocart’s raster reprojection engine to reproject e.g. the red and green Wagner VI image to an equal-area projection and run the pixel-counting script on the result, right?

That should work, if I understand your procedure correctly.

Yeah, Capek probably belongs in Geocart. I guess. I mean, there are so many metrics, though. How to decide which are better?

Best,
— daan

[Atarimaster]

That should work, if I understand your procedure correctly.

Apparently, it does.
I just tried that on Kavraiskiy VII, mainly because that’s I think the only projection in Capek’s list where Q "by cartometry" and "by computer" have exactly the same value, namley 82. I ended up with 81.5 – the deviation by 0.5 can easily explained by the inaccuarcy that is immanent in my approach (e.g. by interpolated pixels that are neither red nor green).

Yeah, Capek probably belongs in Geocart. I guess. I mean, there are so many metrics, though. How to decide which are better?

Good question. I honestly don’t know.
I think there should be at least more than one metric. Offering just one might lead the misunderstanding that there only is one, or at least that this is the metric recommended by you, the author of Geocart.

I chose Capek’s Q in my research because it’s the only metric (that I know of) that I can easily use on all projections that Geocart offers.
It’s also quite handy because it results in a single value that takes both angular and areal distortions into account – although that’s also, in a way, its greatest flaw.
In my opinion, it’s making things a bit too easy. Moreover, the choice of what is deemed an "acceptable" distortion, is somewhat arbitrary, even if there might be good reasons to define 2ωmax = 40°and Kmax = 1.5 like Capek did in his survey.

Kind regards,
Tobias

[daan]

P.S.: Due to my cartometry, which as I’ve said seems to be quite accurate for equal-area projections, the Q of van Leeuwen projection is 91.7 and thus, higher than for any projection that was examined by Capek.

No surprise there. Those projections’ development was much more about achieving their topological characteristics. How to reduce distortion within that format is not a well solved problem.

P.P.S.: Among the interrupted projections Capek lists a projection called BSAM. Yet the only projection bearing this name that I know of is an uninterrupted cylindric projection. So which projection is he referring to?

I’m not sure, offhand. My books are boxed up right now. As is fairly common among Russian projections named for the institutes that developed them, probably several exist, and calling one BSAM doesn’t disambiguate. I think I took my appellation from Maling. If you prod me about this again in a few months I should be able to cough up a more satisfactory answer.

Best regards,
— daan

[Atarimaster]

P.S.: Due to my cartometry, which as I’ve said seems to be quite accurate for equal-area projections, the Q of van Leeuwen projection is 91.7 and thus, higher than for any projection that was examined by Capek.

No surprise there. Those projections’ development was much more about achieving their topological characteristics. How to reduce distortion within that format is not a well solved problem.

Actually, I think it is surprising.
Because on Q, a higher value is better, the upper limit is 100. Thus, the answer to Capek’s title of that paper "Which is the best projection for the world map?" would be: "The van Leeuwen projection" – at least among those, that were examined by me and Capek himself.
When I say that this is "surprising", I refer to the circumstance that … well, if there is one map projection where probably everyone will notice that it’s distorted, it’s the van Leeuwen projection. It is of course not surprising anymore if you know about the distribution of distortions in the van Leeuwen and how Capek defines Q.

On a side note, I don’t think that Capek really thought that his Q is a way to find the "best" map projection. I guess he chose that title because it makes a catchy headline.

If you prod me about this again in a few months I should be able to cough up a more satisfactory answer.

I’ll try to remember that.

Kind regards,
Tobias

[RogerOwens]

One way of rating a projection by both angular and areal distortion would be to look at a problem caused by both kinds of distortion.

Something that I don't like about both areal reduction and angular distortion is scale-reduction. Especially when that scale becomes less than a reference scale that is the scale along the map's limiting-dimension based on the available display-space (...which could often be taken to be the equator, for a map in equatorial-aspect).

Another possible choice of reference scale could be the map's average-scale. (...the square root of the result of dividing the map's area by the Earth's area.)

(In previous posts here, I called that "absolute-compression", as opposed to "relative-compression" that compares min and max scales at a point)

For example, though the Sinusoidal doesn't reduce any region's share of the Earth's area, it drastically reduces min scale in places of high latitude and longitude.

By "absolute-compression" at a point, I mean the min scale at that point, divided by the reference-scale, defined only for values less than unity (...so a smaller number means worse absolute compression.)

Having a smaller number meaning a worse distortion can be justified by saying that "absolute-compression" is short for "factor by which there's absolute compression".

That merit-measure that I propose here, absolute-compression could be stated for a map in several ways, of course:

1. The smallest value that it has on the map.
2. Its mean on the map
3. Its RMS on the map.

I'd be a bit hesitant to rate by RMS though, because some of the most distorted places tend to be some of the least-looked-at places.

So, in that way, by looking at a problem shared by areal and angular distortion, a map could be rated according to one number without making an arbitrary choice about what amount of angular or areal distortion is acceptable.

Michael Ossipoff

[RogerOwens]

I've edited my recent post of today, to add this:

Having a smaller number meaning a worse distortion can be justified by saying that "absolute-compression" is short for "factor by which there's absolute compression".

That merit-measure that I propose here, absolute-compression could be stated for a map in several ways, of course:

1. The smallest value that it has on the map.
2. Its mean on the map
3. Its RMS on the map.

I'd be a bit hesitant to rate by RMS though, because some of the most distorted places tend to be some of the least-looked-at places.

Because I define absolute compression factor as 1, for any point at which min scale isn't less than the reference scale (and where therefore there's no absolute compression), then of course, when determining the average or RMS absolute compression factor on the map, 1 should be recorded and counted for any place where min scale isn't less than the reference-scale.

Michael Ossipoff

[RogerOwens]

I should add a few comments:

I mentioned several ways of rating a map by Absolute Compression Factor (ACF):

1. The smallest ACF value that any point on the map has.
2. Mean ACF
3. RMS ACF

But, in regards to measures of ACF, what would influence my choice would be the lowest value that ACF has, among the regions that are of interest to me.

But of course some measure of low distortion (such as Q, or those measures of ACF) isn't the only consideration in choosing a projection.

For example, though Mollweide doesn't do as well by ACF as some other equal-area maps do, Mollweide has a realistic and aesthetic globelike appearance, with its elliptical shape, and has the topological accuracy and realism of a point pole.

And it's pseudocylindrical. The elliptical equal-area pseuocylindrical.

For general purposes, Mollweide is the Equal-Area map that I'd usually choose. Others have felt that way too, judging by Mollweide's great popularity during the 19th and 20th centuries, and right up to just before the compromise-map fashion took over and equal-area maps went out of fashion.

Michael Ossipoff

[RogerOwens]

Let me just add that, for a reference-scale, I prefer the map's equatorial scale to its average-scale, because ACF, by the equatorial-scale as a reference-scale, registers and notices a low ACF caused by the map being small by the standard of the map's area divided by the square of its equator-length.

...which I call "specific area".

Michael Ossipoff