Eckert IV vs. Eckert VI

[Atarimaster]

Hello,

I’ve got a question regarding Eckert IV vs. Eckert VI.

Both Eckert himself (in "Die Kartenwissenschaft") as well as Wagner (in "Kartographische Netzentwürfe") claim that while Eckert IV has better distortion values (mathematically speaking), Eckert VI is aesthetically more pleasing because the shape of the continents seem less distorted to the eye in comparison to the looks on a globe.

I’ve got to admit: I fail to see that, at least generally speaking. Of course, Africa (horizontally) and Greenland (vertically) are stretched less on the Eckert VI. But on the Eckert IV, North America (except Alaska), Australia and most of Asia seem, in my opinion, closer to the way they look on a globe.
But who am I to disagree with experts like Eckert and Wagner?

So, am I somehow seeing things wrong?
Or is it just a matter of where you look exactly?
Or purely a question of taste?

Regards,
Tobias

[daan]

It seems to me it’s a matter of what you care to look at. Australia and South America are obviously better on Eckert IV, and so is temperate North America. The arctic stretching on Eckert IV is considerably worse. Maybe that’s part of what Mssrs. Eckert and Wagner fixate on, since it makes North America and Asia look a lot wider than they are. Or perhaps they don’t find the more pronounced shearing on Eckert VI objectionable on the theory that it resembles the shearing toward the limbs of a perspective view, and therefore the brain compensates easily. But who knows.

Eckert IV’s basic shape clearly won out; it’s a lot closer to Robinson’s, and people seem to strongly favor the elliptic curves (such as for Eckert IV) over sinusoidal (such as for Eckert VI). In support of the latter claim, see a recent paper I contributed to, User preferences for world map projections.

So no, I don’t think you’re looking at things wrongly at all. I suspect Wagner and Eckert’s judgments there were more a product of time and place than of objective sentiment. Or, as you put it, of taste.

Best,
— daan

[Atarimaster]

Okay, thank you, I’m relieved to hear that. I seriously was beginning to ask myself why I couldn’t see it…

Don’t get me wrong, I really like projections with sinusoidal curves, but mostly because they are rarely to be seen nowadays and therefore, look interesting (just like anything that’s unusual will raise your interest). On the other hand, I always felt that (most) projections with elliptic curves convey a more "accurate" idea of how the earth looks like (that is, as far as any projection is capable of doing).

For the sake of completeness, I’d like to add the Eckert and Wagner quotes that I was referring to, and try to (loosely) translate them.
Please note that both of them use terms like "Ellipsenprojektion" and "Sinuslinienprojektion" – I guess you can understand those words without translation –, but in my translation I’ve inserted "Eckert IV" and "Eckert VI" instead.


Eckert’s "Die Kartenwissenschaft" is available at archive.org, the quote is at page 166 (printed page number) resp. 188 (page number within the PDF document):

W. Behrmann führt in seiner S. 164, Anm. 1 genannten Abhandlung aus, daß die mittlere Maximalwinkelvezerrung (…) in Eckerts Sinuslinienprojektion 32° l9´ beträgt, dagegen in Eckerts Ellipsenkonstruktion nur 27° 34´; er schließt daher (S. 28): „Es ist somit von den Eckertschen Projektionen die Ellipsenprojektion die beste und nicht, wie er annimmt, der Entwurf mit den Sinuskurven.“ Das stimmt mathematisch, aber geographisch kann man auch anderer Ansicht sein. Wohl war mir bewußt, wenn ich es auch noch nicht besonders zum schriftlichen Ausdruck gebracht hatte, daß meine Ellipsenprojektion die geringste Winkelverzerrung hat, und dennoch habe ich mich für die Sinuslinienprojektion als die geeignetere von beiden entschieden, weil sie die Kontinente, d. h. die figürliche Ähnlichkeit nicht so verzerrt wie die Ellipsenprojektion. Der Vergleich mit dem Globusbilde ist hierbei ein guter Korrektor.

Translation:

W. Behrmann notes that the mean maximum angular distortion in Eckert VI is 32° l9´, while it is merely 27° 34´ in Eckert IV, and concludes:
"Thus, among Eckert’s projections the Eckert IV is the best one, not Eckert VI as he [Eckert] assumes."
Mathematically, that’s correct, but geographically you can have a different view. Although I didn’t mention it in writing, I was well aware that the Eckert IV has a lower angular distortion, nonetheless I decided to choose Eckert VI to be the more suitable projection, because it doesn’t distort the shapes of the continents as much as the Eckert IV. The comparison with the view of a globe is a good corrector.

Wagner’s "Kartographische Netzentwürfe" doesn’t seem to be avaiable online, so I’ve scanned page 232 (you can see that I’ve marked the section with a pencil):

Den geringeren Durchschnittsverzerrungswert hat der Ellipsenentwurf, und doch zeigt der Sinuslinienentwurf in bezug auf die Kontinente ein gefälligeres Bild. Allerdings sind die in der Umrandung auftretenden Ecken angegriffen worden. Doch ist die Frage der Umrandung bei einer Erdkarte durchaus eine Frage des persönlichen Empfindens. Wie man sieht, spielen hier sogar nicht mehr exakt meßbare Belange psychologischer Natur herein.

Translation:

Eckert IV has a lower mean angular distortion but still, the Eckert VI shows a more pleasing image regarding the continents. Certainly the corners of the neatlines have been criticized. But the question of neatlines is surely a matter of personal opinion. As you can see, even psychological aspects which aren’t quantifiable are involved here.