Questions regarding Györffy’s paper

[daan]

The American polyconic is not terribly problematic numerically, but the formulæ are singular at the poles and along the equator. If you’re not handling those specially, then you’ll get NaNs.

— daan

[Atarimaster]

The American polyconic is not terribly problematic numerically, but the formulæ are singular at the poles and along the equator. If you’re not handling those specially, then you’ll get NaNs.

Aaah! A valueable hint!
I already looked at the d3 function for the original polyconic, but I must have overlooked the first line:

Code: Select all

function polyconicRaw(lambda, phi) {
  if (abs(phi) < epsilon) return [lambda, 0];
  (…)

with

Code: Select all

var epsilon = 1e-6;
var epsilon2 = 1e-12;

(two of the predefined variables in the d3 scripts).

I added that line to the umbezifferte polyconic (but using epsilon2) and got a result that almost looks right, testing one of the parametrizations that look a lot like Ginzburg VI:

umbezifferte-polyconic.png (139.59 KiB) Viewed 2196 times

As you can see, there’s still a little problem, but at least it stays outside of the map itself. With epsilon, there were also some visible flaws along the equator within the map. So, here’s one of the parametrizations that are very much like the Wagners shown above (this time, I just deleted the erroneous lines in an image editor):

umbezifferte-polyconic-2.png (61.35 KiB) Viewed 2196 times

So, thanks a lot for pushing my in the right direction!
Kind regards,
Tobias

[Atarimaster]

First off, a correction.
Above, I wrote:

However, I was able to transcribe the Wagner III variants.
Here’s the one optimized for Airy:
(…)
… and the one for Airy-Kavrayskiy:
(…)

The second image was wrong!
I got confused with the various examples that Frančula is presenting.
The first image was right, namely being the best possible parametrization for Wagner III according to the Airy criterion, using the parameters C_p = 2.07, C_e = 2.31, ϕ_p = 63°, with C_p being the ratio of the main axes, C_e the length of the equator towards the length of the pole line and ϕ₀ of course being the standard parallel.

The second image however was the best possible parametrization according to the Airy criterion if ϕ₀ = 40° (listed because it’s a frequently used standard parallel), which of course demands to adjust the other two parameters (C_p = 2.10, C_e = 2.35).

The correct best possible parametrization according to the Airy-Kavrayskiy should look like this:

wagner3__0.69-195-44.png (130.52 KiB) Viewed 2178 times

Frančula is not showing an image for this one, he is only giving the parameters: C_p = 1.95, C_e = 1.45, ϕ_p = 44°.


Now, I’ve got a question.
Of course both C_p and ϕ₀ change the aspect ratio of the projection. So why does Frančula use them both? If I get this correctly, the answer to that is given in Geocart’s manual:

Areal inflation/deflation (…) are relative to whatever area is considered to be correct. Normally that would be a region whose area measurement matches the nominal scale’s square area. (…) But since the choice of nominal region is in some sense arbitrary (because the nominal scale itself is arbitrary), merely changing the choice of the nominal region will change which regions of the map are deemed more and less distorted.

So in order to get the best possible distortion value, you have to set the nominal scale by using the standard parallel in order to know how much a region is in- or deflated. Then, you can furtherly optimize the distortion values by simply stretching or compressing the projection using C_p.
So you can end up with two projections that look absolutely identical by using appropriate values for C_p and ϕ₀, but would end up with different distortion values calculated by the Airy or Airy-Kavrayskiy criterion.

Is this about right or did I get something wrong here?


Finally, there’s something that I forgot to mention before:
Frančula’s (German) paper can be requested at researchgate.net. I actually did that in December but got no reply, that’s why I resorted to the used book stores.

[daan]

Tobias,

Your interpretation looks right to me: Those metrics change in response to nominal scale change.

Dr. Frančula is still “in business” as far as I know. Contacting him directly (not through researchgate.net) might yield results, if anyone were to try. I might.

Thanks, and cheers,
— daan

[Atarimaster]

Coming back to that matter after a “short” break…

I got help from a nice guy called Peter Denner who contacted me a while ago regarding an error on my website…
He improved my d3 scripts and expanded them (that is a story for another day). Now I can render all 14 projections that were presented by Frančula. I’ve still got to look into a few things, e.g. I’ve got to check three of them that I’ve called approximations for now if they are really close enough to the original.

Here’s a list of the Frančula projections incl. the mean overall distortion values as noted by Frančula.

[daan]

Thanks for persevering, Tobias.

Can you explain more about the “approximations”? What about them are approximate?

Cheers,
— daan

[Atarimaster]

Can you explain more about the “approximations”? What about them are approximate?

Yes, I can.
Frančula used (like mentioned above) both the standard parallel ϕ₀ and the variable C_p (which describes the aspect ratio of equator and central meridian) to set the aspect ratio of the finished projection.
In the formula that renders II, VII and XI, there is no standard parallel. But again, there are two values that influence the aspect ratio, just like Wagner did it when he introduced the Wagner IX: He started out with p, which was set to 2 (= equator is twice as long as the central meridian), developed the formula using this value, and in the end, he said that you might want to add a horizontal compression "to achieve a better distribution of distortions" by multiplying the x-values with a factor a < 1.
While Geocart is rendering the Wagner IX without using a, Canters obviously referred to the variant with a = 0.88 (which was presented by Wagner as an example) in Small-scale Map Projection Design.


While p also influences the scale of the projection, p does not.
Here’s a link to a "work in progress" example:
https://map-projections.net/d3-customiz ... ner369.php
That page is not finished, e.g. links don’t work etc.
When you use the sliders on the left you’ll see how p and a work differently.

The bottom line is:
Since Frančula uses a standard parallel, but the renditions of the projections II, VII and XI do not, I had to approximate them visually.

By the way, on the unfinshed page you also can see the thing that I called "a story for another day": Umbeziffern applied to van der Grinten IV and Nicolosi – the "telophasic" (as I call it) variant, that’s this one.
That’s the part that Peter Denner wrote.

[Atarimaster]

Ach, it’s late over here and obviously, my brain has already gone to bed…

While p also influences the scale of the projection, p does not.

Except that I wrote p twice, it actually is:
While p applies an affine transformation, a does not.

[Atarimaster]

I’ve got another question regarding Györffy’s document or more specifically the E_K values listed in Table 1 (page 7 of the PDF).

These values are calculated by the Airy-Kavrayskiy criterion, right?
As I’ve mentioned above, Frančula uses the same criterion, called E_AK there. He also lists the values of some well-known projections (page 67 in his paper). Both of them inspect the area between 85° N and 85° S. However, Györffy’s and Frančula’s values differ. For example:

Sinusoidal – Frančula: 0.4701 / Györffy: 0.66474
Mollweide – F: 0.3774 / G: 0.53375
Aitoff – F: 0.3690 / G: 0.52187
Kavrayskiy VII – F: 0.2614 / G: 0.36930
Winkel Tripel – F: 0.2597 / G: 0.36699


So each time, Györffy’s value is about 1.41 times higher.
Do I get something wrong here or how can this be explained?

[Atarimaster]

Can you explain more about the “approximations”? What about them are approximate?

Nothing anymore!
I was able to convert the standard parallel to the horizontal compression factor.
And the answer was there, right before my eyes – namely in the formulae Frančula provided. Don’t know how I managed to overlook it all the time.

And:

So each time, Györffy’s value is about 1.41 times higher.
Do I get something wrong here or how can this be explained?

It was Peter Denner again who found the answer:
Györffy’s E_K is Francula’s E_AK × sqrt(2)

I don’t know what’s the reason for this. Probably there is one, but to me, a layman, it seems quite confusing. We’ve already got lots of different metrics to obtain the “overall distortion”, why furtherly complicate things by noting the same criterion in two different ways?