The Wagner Projections (Part 3): Umbeziffern – The Wagner Transformation Method
Part 3 of this article series was supposed to be about the Wagner variants created by Frank Canters and Dr. Rolf Böhm. For a better comprehension of these variants it occured to me that before that, it might be better to explain the method that Wagner used to create his nine projections: Das Umbeziffern – a term which might be translated as »renumbering« or »coordinate transformation« or maybe »reassigning of values«. In this article however, I’ll just stick to the german word.
Karlheinz Wagner particularised Umbeziffern in 1949 ^{[1]}, Frank Canters provided a nice summary ^{[2]}:
The transformation method is based on a very simple idea, but provides a powerful mechanism for the development of new map projections. First a wellchosen part of the graticule of an existing projection, bounded by an upper an a lower parallel, and a left and right meridian, is selected. The entire area to be represented is mapped onto this part of the graticule by redefinition of the longitude and latitude values of each meridian and parallel (Umbeziffern). Then the graticule is enlarged to the original scale of the parent projection. Restoration of the original scale may be followed by an affine transformation in the x and ydirection. This permits control of the ratio of the axes of the projection.
Let’s illustrate the transformation method using the construction of Wagner VII as example.
Umbeziffern, step by step
As parent projection, Wagner chose the equatorial aspect of the azimuthal equalarea projection; using the part between 65° N/S and 60° E/W for the new projection.
The area to be represented – in this case, it’s the entire earth – is mapped onto this part, then the graticule is enlarged to the original scale of the parent projection:
By an affine transformation, we control the ratio of the axes. The equivalence of the parent projection is preserved. The Wagner VII projection is complete:
So up to this point, the new projection is determined by the following parameters:
 The parent projection;
 the bounding parallel;
 the bounding meridian;
 and the scaling factor to control the ratio of the axes.
Within this article, we’ll confine ourselves to Wagner VII and VIII, which both are based on the equatorial azimuthal equalarea projection. So we’ll drop parameter #1 and take it as a permanent feature. That’ll leave us, for the moment, with three parameters.
Before continuing, I’d like to point out that the chosen bounding parallel in effect determines the length of the pole line of our new projection, while the bounding meridian determines the curvature of the parallels. To clarify, let’s look again at the part of the parent projection, but this time bounded by 80° North/South, and bounded by 90° East/West:
Parent projection truncated at 80° N/S and 60° E/W (left);
truncated at 65° N/S, 90° E/W (right).
It’s quite obvious that higher degree values of the bounding parallel will result in a shorter pole line of the new projection, and higher degree values of the bounding meridian will result in a stronger curvature of the parallels.
Let’s go on:
As I’ve said, Wagner VII is an equalarea projection, just like its parent projection. So Wagner added
two further parameters to control areal distortion:
Man kann nun vorschreiben, dass beim Parallelkreis φ_{1} die Flächenverzerrung nicht größer als S_{1} sein soll (…)
You can prescribe that at the parallel φ_{1} the areal distortion should not exceed S_{1} (…)
For the projection nowadays known as Wagner VIII, Wagner set φ_{1} = 60 and S_{1} = 1.2, i.e. an areal inflation of 20% at 60° N/S. These values were chosen because between ± 60° »you roughly get the populated parts of the earth« and »an areal inflation of 20% for the specified area still seems sustainable«.
So the final list now contains five configuration parameters – and this time, we add the notations usually used in cartographic literature:
 The bounding parallel: ψ_{1}
 The bounding meridian: λ_{1}
 The reference latitude for the areal distortion: φ_{1}
 The amount of areal inflation at φ_{1}: S_{1}
 The aspect of the axes: p
And now, you’re surely beginning to guess what Messrs. Canters and Böhm did to build their Wagner variants: Correct,
they started to twiddle with exactly these five parameters. I’m going to review their result in the pending fourth part
of the article series.
But before that, I’ve still got to mention a thing or two…
Notation
In cartographic literature, you’ll often find the term Wagner constants (or something like that) – because the formulae that Wagner used for his projection, contain a few numerical values, like for Wagner VIII:
I’ve highlighted the constants I’m talking about using red print.
But how did Wagner come up with exactly these numerical values?
Well, he used his five input parameters mentioned above, processed them through a bunch of
formulae, and ended up with some numerical values, that he inserted in the final formula for
convenience of the reader. Of course, with these numerical values you can only
generate Wagner VIII. But instead, you could also write a generalized version of the formula:
Now, you just have to insert appropriate values for m_{1}, m_{2}, C_{x}, C_{y} and n – and you’ll have a Wagner variant, just like those presented by Canters and Böhm. So how do you get some »appropriate values«? – You can calculate them from the five configuration parameters mentioned above, like e.g. Dr. Böhm showed in his paper. That document is in german, however if you read formulae 7 to 12 on page 4/5, I think you’ll get the idea even if you don’t understand German.
And there’s another way to get the values.
But hang on a bit, I’ll come back to that in a moment.
Before that, I’d like to mention that Dr. Böhm proposed a very convenient notation to identify Wagner variations:
A simple list of the 5 convenient parameters, separated by a dash. The first three parameters are written
in their usual values in degree. For #4 and #5 he decided to use percental values, so that an areal inflation
of 1.2 is written as 20, and if the equator shall be twice as long as the central meridian,
it is noted with 200.
Thus, Dr. Böhm’s notation for Wagner VII is 6560600200 and for Wagner VIII, 65606020200.
I think that’s a great idea, because for one thing the values are easier to memorize than the constants (0.92118 etc). But mainly because the values are directly tied to the parent projection, and if you keep the stepbystep images shown above in mind, you might be able to picture (at least approximately) a configuration like e.g. 65846025200, even before you have actually seen it.
However, there’s a little catch: Different notations might result in identical projections. So this notation is unique but not bijective, i.e. there’s no onetoone relation between the noted values and the actual projection. For example: As I’ve said before, Wagner VIII shows an areal inflation of S = 1.2 at 60°. That corresponds to S = 1.89 at 80°, so instead of writing 65606020200, you could also write 65608089200. One could avoid that by agreeing to always note the areal distortion at 60° – so Wagner VIII would be 656020200 – or the other way round, to always note the latitude at which the areal distortion is exactly 1.2 (656060200).
Dr. Böhm did point this out in his paper, but decided to maintain all five parameters, for reasons of clarity and comprehensibility. I agree with this decision but I strongly suggest that whoever may adopt this notation should always use 60 for the third parameter.
Render your own Wagner variations – The WVG
I hope that you’re now curios to get to know how this Wagner variations might look. Because I might have just the right thing for you: The Wagner Variations Generator (WVG). There you can modify four of the five configuration parameters, hit the »Render my projection« button, and an image of the resulting projection will be shown.
Apart from entering some values, you can select a predefined projection which can be rendered using Wagner’s formula, so you can select one of them as a starting point for your own experiments.
Furthermore, there are some image options which don’t refer to the projection in itself but the generated image:
You can choose how the continents are displayed (as a black silhouette, as outlines, with colored countries, or not at all),
the step range of the graticule and the size of the image. To compare your new projection to existing ones,
you can set a background image showing one of the cylindrical, pseudocylindrical und lenticular projections that
are offered here on this website.
You can download the resulting projection as SVG file.
But what am I talking about?
Head over to the Wagner Variations Generator and see for yourself…
In case you’re wondering why you can modify only four of five parameters and not all of them:
Since I suggested that the reference latitude for the areal inflation φ_{1} should always be 60°, I felt
it’s appropriate to stick to that value. Moreover, it helps to avoid errors: With φ_{1} = 60 the
maximum value for areal inflation is 99.999%. Value above that won’t generate an projection image (because an arccos(x) function
will be fed with an illegal value). With φ_{1} = 80 however, the maximum value is 475.877.
Every specific value of φ_{1} leads to a different maximum of S_{1}.
So in order to avoid the frustrating situation to enter values that can’t generate a projection, I fixated the 60 for the latitude and set
the max. value of distortion to 99.999.
A few more words…
Wagner’s Umbeziffern is a powerful tool to create projections according to your needs and preferences, yet it has its limits: You’ll never be able to escape the general characteristics of the parent powerful. For example, you can’t change the spacing of the meridians, nor will you ever generate such curve shapes as you can using Hufnagel’s system (see corresponding article or the interactive Hufnagel tool on mapthematics.com).
Finally, I’d like to stress that this article (and the WVG) covers Wagner VII and VIII only.
But the same holds true for all nine Wagner projections, albeit with fewer options:
Wagner I/II and IV/V are based on pseudocylindric projections and therefore, always have straight parallels –
so the curvature of the parallels isn’t available there.
Wagner IX has equally spaced parallels (at the equator), so while you could modify the curvature as well as the pole line length,
you can’t change the areal inflation as well. Finally, Wagner III and VI are pseudocylindricals and have equally spaced parallels, so
there’s the pole line length left to modify.
And to be honest, there already are well enough pseudocylindric projections, there’s no need to add some.
And by the way…
did I invite you to check out the WVG? ;)
References

↑
Wagner, Karlheinz:
Kartographische Netzentwürfe.
Leipzig 1949. 
↑
Canters, Frank:
Smallscale Map Projection Design.
London & New York 2002. 
↑
Dr. Rolf Böhm:
Variationen von Weltkartennetzen der WagnerHammerAitoffEntwurfsfamilie
First publication in: Kartographische Nachrichten Nr. 1/2006. Kirschbaum: BonnBad Godesberg.
Quoted from www.boehmwanderkarten.de/archiv/pdf/boehm_kn_2_2006_2015_complete.pdf (german)
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