Wagner VIII and Hufnagel projections

[Atarimaster]

Hello,

I'm just trying Geocart and wow, it's a powerful application!
Probably too powerful for me, i.e. too expensive just for playing a bit with map projections (which is what I do) but nonetheless I think it's worth its price.

So just out of curiosity, is there a specific reason for not including the Wagner VIII projection?
I think it's a very fine projection, well-balanced between the preservation of shapes and sizes, but more importantly Geocart has Wagner I to VII, plus IX (Aitoff-Wagner), so the decision to leave out a single projection from Wagner's line seems difficult to understand to me.


Another question:
In Richard Capek's Which is the Best Projection for the World map? (PDF file) there are several "Hufnagel" projections mentioned – to be accurate, six projections which rank among the top 30. I've tried to find images showing the Hufnagel projections and I've failed to find a single image or even further information about them so probably you could regard them as unimportant. Nonetheless they seem to have their virtues. Maybe it would be nice to include them into a future version of Geocart?

Of course I understand that it's neither possible nor desirable to include each and every map projection there is.
I just would like to know whether it was an decision not to include the Hufnagels (i.e. "Not important enough", "Not distinctive enough compared to existing projections" etc.) or if it's just a matter of not finding the formulas you need to include them?

Kind regards,
Tobias Jung

[daan]

Hello Tobias, and welcome to the forums.

Wagner VIII (as well as IX) isn’t in Geocart simply because I have not had convenient access to the generating formulæ. The same goes for the Hufnagel series. They are from German papers presented in 1988–89. Since I’m not in academia, I don’t have convenient access to them, and no one has ever mentioned them to me before. Nor do they appear in other texts. Certainly I am not opposed to adding them!

Happy mapping.
— daan

[Atarimaster]

Hello dann,

thanks for replying.

Wagner VIII (as well as IX) isn’t in Geocart simply because I have not had convenient access to the generating formulæ.

Umm, as far as I know (CMIIW) Aitoff-Wagner is Wagner IX.
Regarding Wagner VIII, Dr. Rolf Böhm is showing Wagner formulas on hins web site (the site is in german but this page is in english). I don't know if that's sufficient to include it in Geocart.

The same goes for the Hufnagel series. They are from German papers presented in 1988–89.

Well, I just received the issue of the "Kartographische Nachrichten" containing Hufnagel's article (I ordered it from a antiquarian bookshop), but sadly, the way I understand german copyright laws, it's not legal to scan the article and send it to you. I will try to find out if there is any legal way to supply you with his formulas and tables, but this might take some time… our copyright laws are very complicated and I'm not a lawyer…

Kind regards,
Tobias

[daan]

Hello Tobias.

Snyder agrees Wagner IX is the same projection as Aitoff-Wagner. I will add that synonym to Geocart. Thanks for pointing that out. I see that Dr. Böhm’s site notes Hammer-Wagner as a synonym for Wagner VII. I will also add that.

Speaking of Dr. Böhm’s site, it looks like there is sufficient information for me to proceed with Wagner VIII. However, there are several variants (and that is true of Wagner VII as well). I will consider how to present these. Thanks for directing me to that site.

I checked ABEBooks.com for the Hufnagel issue of Kartographische Nachrichten, but none were found. If it turns out you are able to transmit the formulæ for Hufnagel’s projections, then many thanks in advance.

Thanks also, by the way, for the kind words about Geocart in your original posting.

Best,
— daan

[Atarimaster]

Speaking of Dr. Böhm’s site, it looks like there is sufficient information for me to proceed with Wagner VIII. However, there are several variants (and that is true of Wagner VII as well). I will consider how to present these. Thanks for directing me to that site.

He's describing the variants on this page, but that's in german so here is what he says:
Wagner VIII Variant 1 to Wagner VIII Variant 4 are Dr. Böhm's own variants that he presented in Kartographische Nachrichten 1/2006.
The "Canters Optimisations" were published by Frank Canters in Small-scale Map Projection Design, London: Taylor 2002.

By the way, Frank Canters developed some beautiful projections himself (which, I guess, you probably already know), Dr. Böhm is showing them, but that's in german, too. At the end of the page he's providing links to assembler source code to calculate them – I don't know if this is helpful for you.

Thanks also, by the way, for the kind words about Geocart in your original posting.

You're welcome!

[RogerOwens]

Someone mentioned Wagner's extended-pole compromise world maps, and it should be mentioned that, for an extended-pole compromise world map, Wagner IX excellently preserves good shape in both low and middle latitudes, unlike Robinson, Eckert III, or any of the 3 Winkel-Tripel versions.

My next poll will include Wagner IX, in addition to Eckert III, Robinson, Times Winkel, and NGS Winkel...in addition to other maps that aren't extended-pole compromise maps.

Eckert III soundly and solidly beat NGS Winkel in my first poll. My 2nd pole didn't include Eckert III, but Robinson came in #1, with NGS Winkel near bottom. So next I'll find out how all of the extended-pole compromise maps in the paragraph before this one do, in comparison to eachother.

Michael Ossipoff

[Atarimaster]

Wagner IX excellently preserves good shape in both low and middle latitudes, unlike Robinson, Eckert III, or any of the 3 Winkel-Tripel versions.

There are three Winkel Tripel versions?
According to this document, »Winkel chose 50 degrees 28 minutes North & South as the standard parallels« which was adopted by the NGS. Then, there is the Bartholomew version, used in the Times Atlas, with 40° N&S as standard parallels.
What is the third version?

Regarding the Wagner projections (esp. Wagner XII to IX), I think it’s regrettable that they are rarely used. I like them very much, and to quote daan from his interview at cartographicperspectives.org people should be exposed to a diversity of projections so from time to time, I’d like to see one of them instead of Winkel Tripel again and again.
Don’t get me wrong, I think Winkel Tripel is a great projection, too, but it seems to be a bit overused these days.

[RogerOwens]

Hi--

You wrote:

There are three Winkel Tripel versions?
According to this document, »Winkel chose 50 degrees 28 minutes North & South as the standard parallels« which was adopted by the NGS. Then, there is the Bartholomew version, used in the Times Atlas, with 40° N&S as standard parallels.
What is the third version?

The 3rd one is the Oxford Atlas version, with lat 0 as its standard parallel.

I often point out that, while there's practical justification in having a mid-latitude standard parallel, with conformality at the central meridian at some mid-latitude (such as lat 40), a world map looks better when optimized for low-latitude shapes, with conformality at the central meridian at lat 0.

It seems to me that, if the world map will be the only map nearby and readily-accessible, then practical considerations suggest the desirability of mid-latitude shape-optimization. That could be important for good estimation of directions, distances and positions. Mid-latitude optimization would be a best compromise, as well as favoring the most populated socially important latitudes.

But what if world map is in an atlas? An atlas has detailed maps of all regions, including continents and smaller regions. ...for close and accurate measurements or estimates of those regions. So the need to optimize mid-latitude shapes seems less compelling. It could still be desirable though, because maybe the region you want to look at is divided between different regional maps, maybe on different pages. Also, sometimes we like the convenience of finding out what we want to find out from the world map, without finding the regional map.

So, in atlases, a case could be made for mid-latitude or low-latitude shape-optimization.

But all the Winkel Tripel versions seem dominated by Wagner IX and Eckert III: They have neither Wagner IX's excellent shapes at both low and middle latitudes, nor Eckert III's great simplicity and consequent easy position-finding.

But Times Winkel and Oxford Winkel are perfectly ok. I just don't like what NGS Winkel does to shapes over most of the Earth's surface.

Regarding the Wagner projections (esp. Wagner XII to IX), I think it’s regrettable that they are rarely used.

Yes, they should be used more. In my poll, Robinson won, because it was the best extended-pole world map in the poll. People evidently prefer the nearly upright continents that extended pole world maps offer. But, in that poll, Robinson's only extended pole rival was NGS Winkel. Very likely people would like Wagner IX, with its better low and middle latitude shapes. Wagner IX would be a good projection for the National Geographic Society to adopt.

I regard Wagner IX as the best extended-pole compromise map--a category of world maps that my poll indicates that people prefer.

and to quote daan from his interview at cartographicperspectives.org people should be exposed to a diversity of projections so from time to time, I’d like to see one of them instead of Winkel Tripel again and again.

That's for sure. And, as daan said, it would be a big aesthetic improvement if a greater variety of world map projections were found in atlases and in National Geographic.

My poll-respondents dislike NGS Winkel.

Michael Ossipoff

[daan]

I wrote:

Speaking of Dr. Böhm’s site, it looks like there is sufficient information for me to proceed with Wagner VIII. However, there are several variants (and that is true of Wagner VII as well). I will consider how to present these. Thanks for directing me to that site.

I programmed Wagner VIII and a variant. I was surprised to see that the original Wagner VIII, along with the “bat-wing” variant I programmed, are not equal-area. Since the constants for Wagner VII do in fact produce a correct Wagner VII in my implementation, I believe it to be a correct implementation. I am not sure if

• The projections are not supposed to be equal-area;
  I programmed them wrong;
  Or, the constants Dr. Böhm gives are incorrect.

If the Wagner VIII is not equal-area, I guess I don’t really see its point. It’s a lot like Wagner VII, even being “mostly” equal-area up to about latitude 60°, having under 20% deviation from equivalence up ’til then. It inflates areas increasingly toward the pole, to no particular benefit from my perspective.

Snyder mentions Wagner VIII but does not explicitly state it’s equal-area or not. I infer from the verbiage he supplies that it might not be. Dr. Böhm’s page states, “General transversal eqal-area azimuthal series,” which caused me to supposed it would be.

Best,
— daan

[daan]

And, indeed, in order for the Wagner VII–VIII series to be equal area, this equality must be satisfied across the entire map:

• a∙b∙m∙m₂∙n∙cos(m₂φ) = cos φ

The relationship requires that m₂ = 1. Hence Wagner VIII is not equal-area; nor are any of the variants in which m₂ ≠ 1.

— daan