Questions regarding Györffy’s paper

[Atarimaster]

I’ve recently mentioned Györffy’s paper on minimum distortion pointed-polar projections. Here are two quotes from that paper:

In addition, some projections were created representing the poles as concave curves (bent towards the equator) by suitable renumbering of Aitoff and ordinary polyconic, with even lower EK values (…) They are ignored in the praxis of cartography because of their appearance.

The shape of the mapped Earth in the renumbered Aitoff and ordinary polyconic (…) suggest that a minimized distortion projection, representing the pole as a line, generates an outline with concave pole lines similar to that mentioned above. The unfamiliar shape of the mapped Earth provokes the neglect of these projections.

Question:
The “renumbered Aitoff” apparently is the Wagner IX (a.k.a. Aitoff-Wagner) – but which projection is the “ordinary polyconic”?
The American Polyconic doesn’t have a pole line – okay, I think he’s talking of a renumbered polyconic here, which could have a pole line; but I can’t remember that Wagner’s Umbeziffern has ever been applied to the American Polyconic. Of course, it’s absolutely possible that someone did and I just didn’t read about it.

And a statement:
Yes, I think it’s true that the Wagner IX hasn’t been used much (or maybe not al all) in the praxis of cartography. But saying that this is because of the concave pole lines seems to be a bold statement to me. I mean, it is an obvious assumption: Winkel-Tripel has been used very often, Wagner IX is quite similar (especially when you compare Winkel-Tripel Bartholomew to the horizontally compressed Wagner IX that has been proposed by Wagner himself) and has rarely or maybe never been used; and their most obvious difference is the pole line. But I think we all know that the most obvious explanation isn’t always the correct explanation.

Moreover, if the concave pole line is what you worry about – well, you can easily get a “pseudo-straight” pole line on Wagner IX (or Wagner VIII, another aphylactic lenticular variant) by cutting off the corners. I’m not fond of this technique, but I’ve seen worse truncations.

So, to me it seems that Györffy is jumping to conclusions here. Or am I missing something?

[daan]

Györffy is definitely jumping to conclusions. There is no research to back up that opinion, so it ought to have been expressed as a conjecture. For my own part, it’s not even a credible conjecture. People are all over the map in their preferences for map projections. Individuals are malleable in that preference. Even a population is malleable over time.

I don’t know what projection he refers to by “ordinary polyconic” subjected to Umbeziffern. Possibly it’s one of the many Russian polyconic modifications from the mid 20th century, such as by Taich or Ginzburg or by the Khar’Kov Engineering Construction Institute, as mentioned by Snyder in Flattening the Earth. I don’t have enough information about those, other than the incomplete set of Ginzburg projections in Geocart.

— daan

[quadibloc]

My guess would be that the "ordinary polyconic" is indeed the American polyconic, but that the operation of "renumbering" it creates the pole line - and that indeed the Russian projections are examples of what he is referring to.

That would mean that "renumbering" is indeed the operation of using the scale along the parallels for a constant times the latitude instead of the actual latitude. In English, the word "renumbering" doesn't suggest that; "scaling" might be better, but maybe the corresponding word in Hungarian would fit nicely.

[Atarimaster]

daan: Thank you for clarifying that!

quadibloc: The word “renumbering” is a literal translation of the German term “Umbeziffern” that Wagner used. For a brief explantion, read the Umbeziffern, step by step section of this document. The images there are a reprodution of the images that Wagner originally presented in 1941 and again in 1949.

Basically, the parallels that previously were 65° N/S (in the parent projection) are now called 90°, the meridians that were 60° E/W are now 180°. So, you given them new numbers – and of course, you have to assign new numbers to all the other parallels/meridians between 0 and 65 N/S or 60 E/W.

I can’t tell if the word “renumbering” makes sense in this context, but the word “Umbeziffern” does.

[quadibloc]

I can’t tell if the word “renumbering” makes sense in this context, but the word “Umbeziffern” does.

Indeed, one could say that the process by which the Russian polyconic projections were made from the American Polyconic is analogous to what was used to create the Wagner VII (or Hammer-Wagner) from the Azimuthal Equal-Area, even though I hadn't been inclined to connect the two (as the Polyconic is more akin to a conventional projection, while the Azimuthal Equal-Area is a true azimuthal, I think of them as being in separate worlds, so even the exact same manipulation applied to both does not feel related).

[Atarimaster]

Question:
The “renumbered Aitoff” apparently is the Wagner IX (a.k.a. Aitoff-Wagner) – but which projection is the “ordinary polyconic”?

I guess I found the answer.

At the paragraph of my first Györffy quote above, he notes a reference to:
Frančula, N. (1971). Die vorteilhaftesten Abbildungen in der Atlaskartographie.

(literal translation: The most advantageous projections in atlas cartography, but I think the way Frančula uses the term "vorteilhaftesten Abbildungen", a better translation might be "Minimum-error projections …")

That sounded interesting, and it was in German, so I looked for it in used book stores on the web, found it and ordered it.
In this paper, Frančula uses Umbeziffern to develop projections with minimum distortions, based on both the Airy and the Airy–Kavrayskiy criterion.

Among his results are variations of Wagner VII and IX, and a few that are labelled "polyconic projection with a pole line", and looking at them, I guess they really could be derived from the American Polyconic. I’d take photos to show them here but I’m not sure about copyright issues… Well, two of them have an outer shape and aspect ratio which are a lot like Ginzburg VI, but with equally spaced parallels along the central meridian. Another one look quite similar to the second Wagner IX variant given below.
There also are umbezifferte projections of sinusoidal and Apian with equally spaced parallels (thus, basically variants of Wagner III and VI).

However I’m pretty sure I can show you his Wagner VII and IX variants. (Not totally sure, though, it’s possible I made some mistakes in converting the values.)

1. Wagner VII, optimized for low overall distortion values as computed by the Airy criterion
(in Geocart’s generalized Wagner, use a = 1.937627, b = 1.25241, m = 0.927184, m2 = 1, n = 0.444444)
Angular distortions:

francula-a7-distortions.png (80.33 KiB) Viewed 2046 times

2. Wagner VII, optimized for the Airy–Kavrayskiy criterion
(Geocart: a = 1.975127, b = 1.328991, m = 0.857167, m2 = 1, n = 0.444444)
Angular distortions:

francula-a28-distortions.png (75.89 KiB) Viewed 2046 times

3. Wagner IX, optimized for the Airy criterion
(no distortion image or Geocart parameters here)


4. Wagner IX, optimized for the Airy–Kavrayskiy criterion
(no distortion image or Geocart parameters here)


By the way, No.1 is not unlike my own Wagner variant labelled "Wagner BCW-E". I chose a different aspect ratio, but pole line length and curvature of parallels are quite similar. I have to say I prefer my own variant – aesthetically (not surprising since it was designed to please my eye) but also regarding the distribution of angular distortions (although I realize that it most likely has worse overall distortion values):

wagner-bcw-a-distortions.png (81.51 KiB) Viewed 2046 times

Finally, I’d like to say that Frančula’s paper is a very interesting read, even apart from the map projections he presents – for me, I doubt that a 1971 paper on map projections contains anything that is new to you guys here.

Kind regards,
Tobias

[daan]

Thanks for the informative write-up, Tobias!

Could you elaborate on what makes Dr. Frančula’s paper a good read? Embarrassingly, I never learned German, so diving into the paper would be a big investment.

— daan

[Atarimaster]

Thanks for the informative write-up, Tobias!

Could you elaborate on what makes Dr. Frančula’s paper a good read? Embarrassingly, I never learned German, so diving into the paper would be a big investment.

You’re welcome!
And, well, it’s a nice introduction to minimum-error projections, measuring the distortions (also mentioning the amount of errors which are introduced by generalization), the Airy and Airy-Kavrayskiy criterion etc.
As I’ve said, nothing that’s spectacular to someone who know all these things, but a good read to acquire knowledge.

Today, I tried to transcribe the formula of the umbezifferte polyconic to the d3 scripts, but …

Bildschirmfoto 2020-01-24 um 18.45.26.png (114.13 KiB) Viewed 2033 times

… so far, I guess there must be something wrong.

[daan]

[Atarimaster]

Still, no luck on the polyconic.
When I (think that I) transcribe the formula as it is printed in the book, I get no result at all, i.e. the script renders an SVG image but instead of coordinates it contains a lot of "NaN". So there’s got to be an error in the calculations but so far I haven’t been able to isolate it.
I get images like the one above when I assume that there’s a typo in the book and wildly change parts to see if that’s the part with the typo.

However, I was able to transcribe the Wagner III variants.
Here’s the one optimized for Airy:

wagner-3-airy.png (44.75 KiB) Viewed 2025 times

… and the one for Airy-Kavrayskiy:

wagner-3-airy-kav.png (165.11 KiB) Viewed 2025 times

One funny thing: When I transcribe the formula exactly as it was in the book, I got this:

wagner-3-airy-err.png (21.27 KiB) Viewed 2025 times

So I merely swapped the definitions of x and y and got it right…
Obviously, I’ve to to do the same thing for the polyconic but also obviously, that’s not the only problem. It’s not that bad because the ones with the best results do look a lot like the Wagner IX variants I’ve shown above, but the Wagners have lower disortion values on both Airy and Airy-Kavrayskiy, so why do I even bother? Well, I just hate it when I can’t get a thing done that can’t be that difficult…

But for today, I’m done.
Kind regards,
Tobias