Was I wrong about Eckert III this whole time?

[justlikeoldtimes]

For the longest time I presumed that the the odd Eckert projections were effectively the average of the plate carrée projection and other projections in existence, if not not by definition or construction, then in practice. (I was not able to find the original 1906 paper that initially presented the six projections, and if I could, I can't read German anyway. So I can't find the initial mathematical development of them.)

Eckert I, if not by original definition, is the apparent mean of plate carrée projection and a degenerate trapezoidal projection where the standard parallels are 0° and 90°. (I nicknamed this the "diamond equidistant projection" since I couldn't find an existing name for it. It should not be confused with the diamond-shaped variant of the collignon projection, which could be constructed in geocart by inputting the correct parameters in the hyperelliptical projection. One is equidistant and the other equal-area.)

Likewise, I believe that Eckert V projection was the average of the plate carrée and the sinusoidal projection.

And until a few days ago, I thought Eckert III could be understood as the blend between the plate carrée map and the Apian II projection, if not by definition then in practice. That is... until I gazed at its distortion visualization in Geocart, and compared it to the blend of the two projections. It's... noticeably different. It surprised me a bit, because they look extremely similar, and for the longest time I found it aesthetically pleasing in some ways.

A few possible explanations:
1. I misunderstand distortion visualizations. It's possible. I wouldn't be interested in map projections if I wasn't a bit interested in mathematics, but my patience with learning math on a higher level is... inconsistent, if were talking about things beyond what I was taught in high school.
2. I misunderstand the Eckert III projection. I see its construction summarized, but not literally translated. Like I said, I don't know how Max Eckert specifically defined it.
3. I misunderstand the Apian II projection. I understood it as an equidistant predecessor of the equal-area Mollweide projection. Maybe I'm unaware of its more specific quirks? Was I thinking of a different, possibly obscure, pseudocylindrical map? If so, what is it?
4. Geocart is wrong? (What do I know?)

As someone who is not a cartographer and might make certain mental shortcuts to understand map projections in a way "that makes sense to me", I would like some clarification.

[PeteD]

Max Eckert rescaled the average of the plate carrée and Apian II such that the resulting projection has the correct total area. If you just do the averaging without the rescaling, then angular distortion should match that of Eckert III but areal distortion won't (unless you use an areal distortion measure that compensates for suboptimal nominal scale).

[Atarimaster]

In addition to PeteD’s reply:

– Open the “Distortion Visualization” dialogue for the Eckert III projection.
– Select “Areal (logarithmic scale …)”.
– As Pivot, enter 0.71314 (that’s the amount of areal inflation at the center of the map, as you can see in Geocart’s Information window).
– As Limit, enter 71.314.
– Hit the OK button.

You’ll see that now the distortion visualization of Eckert III looks exactly like the one of the “handmade blend”.

[justlikeoldtimes]

Thanks. That clears some stuff up. When I get home, I'll also double-check the other five Eckert projections too to see what exactly's going on.

I personally prefer the average without his additional modification. And I guess the map I like was never really Eckert III to begin with.

I'm working on animated gif transitions between a few projections, and I'd rather have the midpoint between plate carrée and Apian II be the true average.

[Atarimaster]

I personally prefer the average without his additional modification. And I guess the map I like was never really Eckert III to begin with.

Well, the only difference between Eckert’s version and the unmodified average is really just the nominal scale.
A 50 × 25 cm map of Eckert’s original has a nominal scale of 1 : 67684300.2029 while it’s 1 : 80132119.6857 on the unmodified average.
Apart from that, the two maps are identical.

Which is why I really have no idea why anyone would care about the “correct total area”. Maybe someone can explain this to me?

[justlikeoldtimes]

Alright, after poking around on Geocart again... I think I understand a bit more now.

The impression I get that the "nominal" aspect of the scale should not affect the end result, (in which the odd Eckerts are indeed the "average" of the said projections), but the nominal scale is directly relevant to how the distortion gets visualized. For some reason, I was under the impression that distortion visualization was not calculated from nominal scale.

I know this is all very basic cartography stuff to all of you, but I'm learning this as I experiment with Geocart; which isn't a bad way to learn I suppose.

[Atarimaster]

For some reason, I was under the impression that distortion visualization was not calculated from nominal scale.

It HAS to be calculated from the scale – how else could you tell that some regions are shown at the "right size" and others are not?
After all, everything on the map is much much smaller than in the real world…

[mapnerd2022]

Does Flexprojector offer both Eckert III and IV?

[Atarimaster]

Does Flexprojector offer both Eckert III and IV?

It offers Eckert IV and an Eckert III approximation (which has disturbing kinks in the meridians near the poles).

[mapnerd2022]

Yes, kinks are always horrible.