Wagner VIII and Hufnagel projections

[Atarimaster]

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.

Wagner VIII indeed is not equal-area, nor is it meant to be.
Wagner called it the "Hammerschen Entwurf mit Pollinie und vorgeschriebener Flächenverzerrung" – I don't know if I can translate that correctly but it might be "flat-pole Hammer projection with required distortion of area".

I guess it was Wagner's intention to enhance the shapes a little bit (esp. Greenland, but e.g. Africa is less elongated compared to Wagner VII) without moving to far away from equal-area and thus, creating something between Wagner VI (equal-area) and Wagner IX. When you look at Wagner I to III and Wagner IV to VI, you see that he always starts with equal area and then moves away from it in two consecutive steps.

In his book "Karthographische Netzentwürfe" he puts some effort to explain why he thinks that there are good reasons for "compromise projections" (neither conformal nor equal area). The way he writes about it makes me think that compromise projections weren't quite well-respected at the time of writing (1949), at least not in Germany, and that he's trying to improve their reputation. So maybe he felt that it's better to stick very close to equal-area in order to gain more acceptance for the projection.

In this day and age, with compromise projections everywhere, I can understand that you might feel that Wagner VIII is obsolete or redundant.
To me, it still is a nice projection, because I generally don't like to move to far away from equal-area… well, that is, unless there is a good reason to do so.

Kind regards,
Tobias

[daan]

I went ahead and put into Geocart all the Wagner VII and VIII variants Dr. Böhm describes on that Web page.

Thank you for commenting on Wagner’s thinking. You may be right. I would still just use Wagner VII, myself; the improvements to Africa and South America are minor. Greenland’s shape does not improve at all on the VIII; it is practically identical in shape to the VII but just too large. If you’re going to depart from equivalence, be bold, I say!


Regards,
— daan Strebe

[Atarimaster]

I went ahead and put into Geocart all the Wagner VII and VIII variants Dr. Böhm describes on that Web page.

That's great!

Greenland’s shape does not improve at all on the VIII; it is practically identical in shape to the VII but just too large.

Looking closely again, I've got to admit: You're right!
Probably it was just the impression that Greenland’s "skullcap" (to me, Greenland always looked a bit like a head shown in profile ) isn't banging as hard on the world's edge in Wagner VIII as it does in Wagner VII made me feel more comfortable with it.

If you’re going to depart from equivalence, be bold, I say!

Good point.
Maybe Wagner just wasn't bold enough – or he was trying to offer a slight compromise because he thought that his contemporaries weren't bold enough…

By the way, in his book Wagner is offering two variants of Wagner IX (Aitoff-Wagner), too:
First, he's giving the formula and tables which result in the "straight" variant (the one that Geocart is generating). But then he's saying that this projection can be compressed horizontally by multiplying the x-values with a factor a and is showing an illustration with a = 0.88 in order shape the projection close to Winkel Tripel.
The latter was labelled "Wagner Y5" by Dr. Böhm; I'm attaching two images of these variants. For the lack of other possibilities, I generated (or rather approximated) the Y5 by simply scaling the IX down horizontally to 88% using an image editor.

And speaking of Winkel Tripel, I've got a question:
Is it really true that the Winkel Tripel was more or less unknown in the USA before it became the NGS standard map in 1998?
I'm asking because my old school atlas which was printed in 1974, Winkel Tripel is used for all world maps (political, physical and thematical maps such as economic, agricultural, languages…) so I always figured it was quite popular already at that time.

Kind regards,
Tobias

[Atarimaster]

I went ahead and put into Geocart all the Wagner VII and VIII variants Dr. Böhm describes on that Web page.

Oh, and can I ask you a favor?
Would you mind uploading images of these variants to Wikimedia Commons?
I only know Dr. Böhm’s low-res images so I'd love to see better versions…

Kind regards,
Tobias

[daan]

Oh, and can I ask you a favor?
Would you mind uploading images of these variants to Wikimedia Commons?
I only know Dr. Böhm’s low-res images so I'd love to see better versions…

Here you go!

• https://commons.wikimedia.org/wiki/File ... ion_SW.JPG
  https://commons.wikimedia.org/wiki/File ... ion_SW.JPG
  https://commons.wikimedia.org/wiki/File ... ion_SW.JPG
  https://commons.wikimedia.org/wiki/File ... ion_SW.JPG
  https://commons.wikimedia.org/wiki/File ... ion_SW.JPG
  https://commons.wikimedia.org/wiki/File ... ion_SW.JPG

And speaking of Winkel Tripel, I've got a question:
Is it really true that the Winkel Tripel was more or less unknown in the USA before it became the NGS standard map in 1998?

I would not say “unknown”, but its use was quite uncommon in the US. I cannot come up wall map examples offhand, but, for example, Preston Everett James and Nelda Davis published a school atlas in 1959 called The Wide World that used the projection for climate maps.

Best for the new year,
— daan

[Atarimaster]

Here you go!

Great, thanks a lot!
But…

https://commons.wikimedia.org/wiki/File ... ion_SW.JPG

This one looks… strange.
Nothing at all like something that Canters would do.

Are you sure that you transferred the formulae given on the website correctly?
If you are, then Dr. Böhm probably has some mistake on his site.

I would not say “unknown”, but its use was quite uncommon in the US. I cannot come up wall map examples offhand, but, for example, Preston Everett James and Nelda Davis published a school atlas in 1959 called The Wide World that used the projection for climate maps.

Thanks again!
And best wishes for 2015 to you, too!
Kind regards,
Tobias

[daan]

https://commons.wikimedia.org/wiki/File ... ion_SW.JPG

This one looks… strange.
Nothing at all like something that Canters would do.

I had the same reaction. However, if we look at the distortion diagrams, the motivation seems a little clearer:

Angular distortion on Canters variant

Canters angular.jpg (55.87 KiB) Viewed 2925 times

Areal distortion on Canters variant

Canters areal.jpg (46.45 KiB) Viewed 2925 times

Notice that the angular distortion along the equator is the same at the outer maximum as it is at the inner maximum. This tactic, which is common in studies of minimal distortion projections, ensures that the maximum distortion across the region of “optimality” is bounded, in contrast with a more typical presentation in which more central locations have no distortion but the extremities distort in an uncontrolled way. The odds we would see the former behavior in a mistaken arrangement are tiny, so I suspect there is nothing wrong with Böhm’s report or my implementation.

In any case, I have Canters’s text; I just need to dig it out of storage to confirm.

Happy new year to you and all!
— daan

[Atarimaster]

Notice that the angular distortion along the equator is the same at the outer maximum as it is at the inner maximum. This tactic, which is common in studies of minimal distortion projections, ensures that the maximum distortion across the region of “optimality” is bounded, in contrast with a more typical presentation in which more central locations have no distortion but the extremities distort in an uncontrolled way. The odds we would see the former behavior in a mistaken arrangement are tiny, so I suspect there is nothing wrong with Böhm’s report or my implementation.

I see your point.

On the other hand, I just checked Böhm’s website again on that matter and realised that he is showing more about Canters’ Wagner-optimisations than I was aware of:
http://www.boehmwanderkarten.de/kartogr ... s_wagner_i

He is showing Canters’ optimised versions of Wagner I, II, VI, VII, VIII and IX; and the Wagner VIII variant does look different…

By the way, it’s interesting to see (if you scroll to the top of that page) that Canters does have two asymmetric of his own – well, I guess you are aware of that but I wasn’t.

Kind regards,
Tobias

[RogerOwens]

Someone here pointed out that compromise projections are everywhere.

Winkel lived in Germany, but it has been suggested that compromise maps' popularity with U.S. publishers is due to the fact that those publishers are following the National Geographic Society (NGS). NGS has, for a long time, been promoting compromise maps.

Someone here said, "If you're going to depart from equal-area, be bold."

Yes, and that's a well-deserved denunciation of "compromise maps"--mediocre, property-less maps.

Eckert III, Apianus II, and Cylindrical-Equidistant aren't compromise maps in that sense. Those 3 projections all have the useful property of linearity.

Michael Ossipoff

[Atarimaster]

This one looks… strange.
Nothing at all like something that Canters would do.

I had the same reaction.

Well, more than a year later…

I finally was able to get a secondhand copy of Canters’ Small-scale Map Projection Design.
And your initial reaction was right – and mine as well –, Canters’ optimised Wagner VIII does look different:

canters-wagner8-optimised.jpg (82.37 KiB) Viewed 2728 times

Kind regards,
Tobias