Györffy's minimum-distortion pseudocylindrical projection

[daan]

I'd be very interested if anyone knows the answer or can provide formulae for Snyder's projections.

The volume is available from the USGS site. I think you’re referring to the projections described in pages 120–131.

Cheers,
— daan

[rschmunk]

I'd be very interested if anyone knows the answer or can provide formulae for Snyder's projections.

Snyder's two minimum-error projections are in his 1985 USGS bulletin report, "Computer-Assisted Map Projection Research".

Flat-pole is equations 5-176, 5-177 and 5-179, using the values of B1, B3 and B5 given on page 128.

Point-pole is equations 5-151 and 5-153, using the values of A1, A3, and A5 given on page 127. However, there seems to be a typo in 5-151, as it says R² when it should just be R.

It's been a year since I implemented these in my code, but I think I compared results to the graphics in Paul Anderson's Gallery and it was a match.

Ooop, I didn't see that another page had been appended to this thread and that daan had already provided a link to Snyder's report.

[PeteD]

Thanks very much to both of you!

[PeteD]

On a related note, what criterion/which metric did John Snyder use when he made his two minimum-error pseundocylindricals?

I don't know, but from the skinny Africa, it looks more Airy than Airy-Kavrayskiy to me.

Now that I've been kindly pointed in the right direction, I can confirm that Snyder did indeed use the Airy criterion. It's equation 5-191 on page 125 of his paper.

[mapnerd2022]

Thanks very much, all of you!

[PeteD]

On a related note, what criterion/which metric did John Snyder use when he made his two minimum-error pseundocylindricals?

I don't know, but from the skinny Africa, it looks more Airy than Airy-Kavrayskiy to me.

Now that I've been kindly pointed in the right direction, I can confirm that Snyder did indeed use the Airy criterion. It's equation 5-191 on page 125 of his paper.

This got me thinking about how Snyder's minimum-error projections would have looked if he'd used the Airy-Kavrayskiy rather than the Airy criterion.

Here's the original Snyder minimum-error point pole projection:

snyder_pp_A.png (154.97 KiB) Viewed 7985 times

and here it is optimized for Kavrayskiy-type angular distortion:

snyder_pp_AK.png (160.33 KiB) Viewed 7985 times

The optimized parameter values are:

A1 = 1.246
A3 = -0.101
A5 = -0.0141

[PeteD]

Here's the original Snyder minimum-error flat pole projection:

snyder_fp_A.png (157.8 KiB) Viewed 7983 times

and here it is optimized for Kavrayskiy-type angular distortion:

snyder_fp_AK.png (157.57 KiB) Viewed 7983 times

The optimized parameter values are:

B1 = 1.257
B3 = 0.211
B5 = 0.155

[PeteD]

Compared to the Airy–Kavrayskiy criterion, the Airy criterion gives more weight to regions of higher distortion, typically the poles (whereas modifying the Airy–Kavrayskiy criterion to take the absolute mean rather than the quadratic mean of the logarithmic distortion values gives less weight to regions of higher distortion).

For the point-pole projection, this explains why switching to the Airy–Kavrayskiy criterion improves most of the globe at the expense of the polar regions, but why isn't this the case for the flat-pole projection? And why do the projections optimized for Kavrayskiy-type angular distortion look so similar to each other when Snyder's original projections look so different from each other?

For the flat-pole projection, Snyder writes that the calculation was limited to 75 degrees N/S, but for the point-pole projection, all he says is that the calculation was carried out at 5-degree intervals. Since the point-pole projection has no pole lines, I presume he could have carried out the calculation all the way up to 90 degrees N/S without computational problems, but whether he did so, or whether he stopped at 85 degrees N/S, is unclear.

In any case, for the flat-pole projection, rather than giving more weight to the polar regions compared to the Airy–Kavrayskiy criterion, Snyder's calculation ignores a large chunk of them.

The optimization for Kavrayskiy-type angular distortion was carried out all the way up to (but not including) 90 degrees N/S for both projections. Unlike with the Airy criterion, the Airy–Kavrayskiy criterion makes this possible even for projections with pole lines because the area of a given latitude band on the globe goes to zero as you move it towards the nearest pole more rapidly than logarithmic distortion values within that latitude band go to infinity for projections with pole lines.

Interestingly, both of the projections optimized for Kavrayskiy-type angular distortion look a lot like a vertically stretched Eckert IV. They have lower overall angular distortion than any other pseudocylindrical equal-area projection that I've tested (and I've tested pretty much all of the well-known ones except for the Tobler hyperelliptical and the Hufnagel projections), just ahead of Eckert IV.

[Atarimaster]

Thank you, Pete, for the interesting comparison and your explanations.

In my opinion, the Airy-Kavrayskiy-optimized projections look much better than Snyder’s originals. It’s a good example to show why nobody seems to use the Airy criterion anymore. However, the vertical stretch is too pronounced for my taste, so I think I prefer Hufnagel projections – Hufnagel 3 or 4 for the point pole, 10 for the flat pole projection –, even if their distortion values may be worse.

Nice work!
Kind regards,
Tobias

[PeteD]

However, the vertical stretch is too pronounced for my taste, so I think I prefer Hufnagel projections – Hufnagel 3 or 4 for the point pole, 10 for the flat pole projection –, even if their distortion values may be worse.

Yes, I've mentioned before that in my opinion, no cylindrical projection and no equal-area projection interrupted along a single meridian or less can ever look good over the whole globe. For these two classes of projection, in my opinion, low overall distortion makes them look less terrible at the poles but quite bad everywhere. Instead, the cylindrical and equal-area projections that subjectively look best are the ones that maximize the area of the globe over which distortion is acceptable at the expense of the regions beyond this area.

On the other hand, in my opinion, this is not the case for pseudocylindrical and lenticular compromise projections, which can look good enough over most of the globe. For these two classes of projection, in my opinion, low overall distortion, at least when measured according to the (weighted) Airy–Kavrayskiy criterion, correlates strongly with what subjectively looks best.