Eisenlohr’s optimal conformal map of the world

[Atarimaster]

Just a short reply, because I’ll have to leave the computer soon…

That's your bias speaking, because Greenland is squashed in so many projections that we develop a poor intuition for what it's supposed to look like.

Aaah, no. Actually, my very first own map projection experiment was pretty much about Greenland’s shape.
But, as you know, on every world map we have to live with a certain amount of distortion somewhere. And considering that, I do think that the projections I named have a comparatively decent Greenland shape IMHO.

Of course, they can’t beat pointed-polar projections in that regard – which is why I also experimented quite a bit with pointed-pole projections which at first glance look very much like pole line projections…

Damn, out of time now…

[Atarimaster]

I wonder why he made the 1st and 2nd projections useless/novelties when from his 3rd projection onwards we have actually useful world maps...

Regrettably, don’t have the original paper in which the six Eckert projections were introduced.

I just found out that Eckert’s original article “Neue Entwürfe für Erdkarten” is accessible on the web.

Of course it’s in German, so it’s probably not helpful for most of you. So here are the parts referring to the “weirdness” of Eckert I and II (translated by DeepL, with a few modifications by myself):

It is an old experience that the construction of geometric figures and thus of maps in general can be done most quickly if only straight lines are needed. On the basis of the coordinate cross with pole lines described above, earth images can be drafted quickly and conveniently with the help of straight lines. Since they are at the same time the simplest, they shall be at the beginning of our projections here.
(...)
As simple and convenient as the projections constructed with straight lines are, they are unfortunately peculiar because of the kinks at the equator. Even if, as a result of the distribution of water and land, only a few mainland areas are affected by the kink in the middle, the projections I and II should only serve basic purposes. For private use, however, especially when it is a question of rapid production of an equal-area map, the second projection is quite suitable, at least much more suitable than Collignon’s equivalent projection.

[mapnerd2022]

I've just downloaded it. I'm going to read it.

[quadibloc]

Oh, an azimuthal projection is polar aspect having kind of “clock-like” time zones works also pretty well in my opinion.

I agree, but I sort of like having North at the top.
And so I split the difference with this projection:

...a version of the conformal conic due to Herschel.

[PeteD]

Oh, an azimuthal projection in polar aspect having kind of “clock-like” time zones works also pretty well in my opinion.

You could use a map like that to make an actual clock by putting the numbers 0 to 23 around the outside (or 1 to 12 twice) and rotating the map such that it completes one revolution every 24 hours. It would show the time everywhere in the world at once. For improved accuracy, you could add a standard minute hand over the top, although you'd have to correct for those pesky half- and quarter-hour time zones.

[Milo]

Oh, an azimuthal projection in polar aspect having kind of “clock-like” time zones works also pretty well in my opinion.

You could use a map like that to make an actual clock by putting the numbers 0 to 23 around the outside (or 1 to 12 twice) and rotating the map such that it completes one revolution every 24 hours. It would show the time everywhere in the world at once.

The thought had crossed my mind. You'd need to arrange the numbers counterclockwise, though, or else center the azimuthal projection on the south pole rather than the north pole.

Though I guess the fact that you have 24 hours instead of 12 means you already don't exactly have a conventional clock arrangement anyway.

[PeteD]

And I don't see why Alaska would be that much more important than, say, Greenland, which is seriously squashed in every pole-line projection I know of, pseudocylindric or lenticular.

Alaska has more than 13 times the population of Greenland, so at least from a human geography perspective, you could say it's more significant than Greenland by the same factor as India is than Egypt, Indonesia is than Mali, Mexico is than Hungary, Australia is than Latvia, Bulgaria is than Malta or Moldova is than Samoa.

Furthermore, the USA is the world's fourth-largest country by size, third-largest by population and largest by nominal GDP and military expenditure, so at least for a political map, it would be nice to depict the USA well, even its sparsely populated areas.

Greenland extends nearly 1400 km further north than Alaska, so in a way, it's more understandable if it's misshapen. I also don't think the fact that Greenland is badly distorted in a given pseudocylindrical projection is a good reason to accept a similarly distorted Alaska when this distortion can obviously be reduced just by curving the parallels a bit.

Starting from a given pseudocylindrical projection, shortening or eliminating the pole line to improve Greenland would result in significantly higher angular deformation at the outer meridians. On the other hand, curving the parallels to improve Alaska wouldn't result in significantly higher angular deformation popping up anywhere else.

I agree that for some specific usages, straight parallels are more appropriate, but in the absence of a particular reason for straight parallels, I would still personally always favour a lenticular projection over a pseudocylindrical.

[PeteD]

You'd need to arrange the numbers counterclockwise, though, or else center the azimuthal projection on the south pole rather than the north pole.

Though I guess the fact that you have 24 hours instead of 12 means you already don't exactly have a conventional clock arrangement anyway.

You can already get anticlockwise clocks and 24-hour analogue clocks, but I can't find any existing 24-hour anticlockwise clocks.

I've heard people wonder why positive angles are conventionally defined as being anticlockwise in right-handed coordinate systems, but I've long suspected it was really the clocks that went the wrong way round

[daan]

I am told that Eisenlohr gives a correction to his formulæ in article XIV here:
Ueber Kartenprojektion
However, I don’t see anything that is obviously a correction. Instead, it seems to be the public debate with August about their respective projections. If anyone can make sense of this, I’d be grateful.

— daan

[daan]

However, I don’t see anything that is obviously a correction.

Of course, I find it right after I post. At least, I think this is it:

Correction

Screen Shot 2022-10-06 at 5.40.48 PM.png (30.84 KiB) Viewed 6588 times

Cheers,
— daan