Wagner III and Wagner VI

[Atarimaster]

Hello,

I’ve got a question regarding Wagner III and Wagner VI.

On Wagner III, Wagner allowed the standard parallel to be arbitrarily chosen by the map maker.
This is mentioned by Snyder in "Flatterning the Earth", (in german) by Dr. Böhm (here), and accordingly in Geocart as well as in G.Projector, the user has the option to enter a standard parallel value.

However, on Wagner VI, this isn’t mentioned by Snyder or Dr. Böhm, and both applications don’t allow to set the standard parallel.
But look at the attached image, photographed from Wagner’s Kartographische Netzentwürfe on the page where he describes his projection 19c (which was later labelled Wagner VI). The framed formula is (I guess) the one that is used by Geocart.
(Dr. Böhm adds: "A simple y=φ would have been sufficient, Mr. Wagner.")

But below that formula, Wagner writes (I’m trying to translate as best as I can)

If you like to compress in direction of the equator (see projection 18c), e.g. in order to map the parallel φ0 equidistant, you have to assign the factor q = cosφ0 / cosψ0 to the x-values.

Note: "Projection 18c" is of course Wagner III


So, am I getting something wrong here – or did Wagner encourage map makers to apply a chosen standard parallel to the Wagner VI projection, just like he did on Wagner III?
If I’m wrong: What is he talking about in the sentence I translated?
And if I’m not wrong: Why isn’t it mentioned anywhere and not available as an option in Geocart?

Kind regards,
Tobias

[daan]

Hello Tobias.

I think your interpretation is correct, but I would argue it’s merely a footnote that applies to any cylindric or pseudocylindric projection, not just Wagner VI. That’s probably why he didn’t include the adjustment in his main formulas. The derivation of the factor varies by projection, of course.

In Geocart you can always stretch or shrink a projection in one or both dimensions by any arbitrary amount. That capability lets you set a chosen parallel to have whatever scale factor you want. You can even figure out what values you should use by noting in the info palette the k scale factor when you have the pointer over the chosen parallel. Multiplying the width of the map by the reciprocal of that value will result in that parallel having constant scale.

Regards,
— daan

[Atarimaster]

Thank you, daan, for both the explanation and the hint regarding Geocart!