Eisenlohr’s optimal conformal map of the world

[quadibloc]

I prefer the compromise ones over the equal versions only on the fact that the equivalent ones make Africa extremely skinny, even more than in the Mollweide. I don't like Africa (and the tropical regions in general) being skinny, as long as it isn't skinny to an equal area Eckert pseudocylindrical level or a Gall-Peters level. Otherwise it doesn't bother me much.

And that's precisely why I think that if an equal-area projection is needed, the Sinusoidal, even though it needs to be interrupted, is preferable, since it doesn't make anything skinny - unlike an Eckert IV.
Of course, there are cases where the Eckert IV is acceptable, but if one is being sensitive to the Mercator giving the "wrong ideas" about the world to young map viewers, then a projection where shapes are obviously grossly distorted isn't the right antidote.
Actually, though, if you're dealing with the little kiddies, there is an obvious map projection option that you should consider using. The Orthographic!

[Atarimaster]

And that's precisely why I think that if an equal-area projection is needed, the Sinusoidal, even though it needs to be interrupted, is preferable, since it doesn't make anything skinny - unlike an Eckert IV.

And if I wanted an an equal-area projection that avoids skinny low latitudes, I’d rather switch to the lenticular class than to interruptions…
Fortunately, there’s an appropriate projection for each preference.

[Atarimaster]

I’d prefer Hufnagel 10 over Eckert IV

Looking at them, they seem very similar, but Hufnagel is slightly closer to cylindric/rectangular?

That just reinforces my idea that beyond a pole line of a certain length (or even sufficiently wide rounded poles, like Tobler hyperelliptical with large exponents), you might as well stop bothering with pseudocylindric projections and just accept the advantages (and disadvantages) that come with an outright cylindrical map.

Good point.
Nonetheless, comparing Hufnagel 10 to the cylindric equal-area projection with standard parallel at 37°(chosen because it roughly has the same aspect ratio as Hufnagel 10) I’d still prefer the pseudocylindrical here, both aesthetically and in terms of the distribution of distortions.

hufnagel-vs-CEA.jpg (77.07 KiB) Viewed 5661 times

[mapnerd2022]

Good idea, you can put it in the «what projections do you favor?» thread. Though Mr.Strebe is the boss here, he is the one to decide that. Do whatever you want!

[PeteD]

However, I dislike pseudocylindrical projections with pole lines as a rule, and don't really see the point.

Well, the point is that they tend to have less distortion than either cylindrical projections or pointed-polar pseudocylindrical projections (except for pointed-polar pseudocylindrical projections such as the Tobler hyperelliptical that exhibit significant east-west stretching at higher latitudes without a pole line).

If you look at Eckert IV, you'll see that it's basically intermediate between Mollweide and Smyth-Craster. The difference is most pronounced in North America, which is less slanted than in the former but more than in the latter, with no real rationale for favoring that exact level of slanting.

I suppose the rationale for having the pole line half the length of the equator is that 1/2 is the "least arbitrary" value between 0 and 1, but you're right that it's not really better than any other intermediate value. To me, if you want to avoid arbitrariness, then a configurable projection such as Wagner IV, where the pole line can have any length between a point and the full length of the equator (and which therefore includes both the Mollweide and the Smyth-Craster as special cases), seems to be a better option than restricting yourself to only the extreme pole line lengths.

Using your example, Eckert IV looks better than the Smyth-Craster in the tropics and especially at the poles, with no significant drawbacks. Between Eckert IV and the Mollweide, the winner is less clear -- Eckert IV has a better Australia and Alaska (although the latter is admittedly still not great), whereas the Mollweide has a better Greenland and a less skinny Africa -- but overall, I still prefer Eckert IV. However, the only equal-area projections interrupted along only one meridian in which all three of Africa, Greenland and Alaska look decent (in equatorial aspect) are lenticular.

Even if you're drawing the graticule, it can still be beneficial for it to have an intuitive shape. The Dymaxion map doesn't stop being horribly confusing just because you're drawing the graticule.

I see your point, but I don't think that curving the parallels a bit is comparable with the Dymaxion.

I'll also note that climate has knock-on effects on numerous other things, and so is relevant to many other types of maps as well.

Perhaps I don't have a good imagination, but the only ones that I can think of are vegetation (or lack thereof) and surface ice/snow (or lack thereof).

Topography? Not really.
Political boundaries? Not really.
Time zones? Not really.
Population density? Only to the extent that it determines the northern limit of densely populated areas -- in the southern hemisphere, the limit can be much better explained by the spread of "civilization" up to the 17th century, as seen by comparing this map by Batholomew showing the "growth of citizenship in commerce and civilized culture" with this map showing modern population density, although it can be argued that "civilization" developed in Europe and the Mediterranean, the Middle East, India and East Asia precisely because these regions are located on the landmass with the greatest east-west extent.

[Milo]

I'll also note that climate has knock-on effects on numerous other things, and so is relevant to many other types of maps as well.

Perhaps I don't have a good imagination, but the only ones that I can think of are vegetation (or lack thereof) and surface ice/snow (or lack thereof).

Well, that's a pretty big one. 99% of the source maps used to demonstrate map projections on this forum or on Daan or Atarimaster's websites are more-or-less natural-color maps (except that they like to recolor the oceans for some reason), where the most notable colors are forest-green, desert-yellow, and glacier-white (and whatever shade of blue gets used for the oceans, and a bit of red in the Australian desert specifically).

Of course, climate is complicated and there are numerous other factors that affect it (and therefore biome), but latitude is a dominant enough one that even when places have a climate different from expected for their latitude, this is important information that should prompt you to ask why, so it's good if the discrepancy is easily visible.

Climate in turn affects not only plant but also animal migration. Consider how many cold-adapted animals have "holarctic" distributions, found across both Eurasia and North America, while their tropical faunas are more separate - largely because the only easy way to travel between the two is across Beringia. Unless you're a bird, in which case crossing the ocean is a lot easier, and parrots, for example, do have a pantropical distribution. Relatively closeby islands like Hokkaido and Honshu can have very different faunas if they're orientated north-south to each other, whereas their faunas are more similar to the mainland Asian environment at the same latitude despite the sea separating them. Marine animals, of course, also care quite a bit how warm the waters are.

Climate also historically affected human migrations. Consider how the Silk Road allowed trade all the way between China and Europe even in medieval times, whereas the Incas and the Aztecs weren't even aware of each other's existence. Or how the Arabs, long used to living in deserts, were able to spread Islam and the Arabic language across the Sahara all the way to Morocco, yet never penetrated far into sub-Saharan Africa. This affects political and cultural affiliations to this day, as well as (you mentioned this one) the spread of technology.

Prevailing winds, another consequence of climate, were extremely important to navigation in the Age of Sail, and although less important nowadays that we have more powerful engines capable of pushing against the wind, modern airliners still make use of them to minimize fuel expenditure.

Latitude is also the easiest coordinate to measure using traditional celestial navigation techniques (for largely the same reason that it affects climate so much: the sun's at a different angle), though that's less relevant now that we have GPS.

When it comes down to it, latitude has at least some bearing on nearly everything. Longitude is much less important, with the main benefit of cylindrical projections being not so much that meridians are straight lines, but rather that all meridians are the same, avoiding bias to certain parts of the world (at least in an east-west direction). Just about the one thing that longitude does matter for is... well, see below

Really, the atmosphere and weather are the reason why our planet even has life to begin with. You can't ignore it.

There are exceptions, of course. Tectonic plates don't care one bit what the sun is doing, so having straight parallels may be less relevant to geological maps. Then again, there are other geological phenomena, such as soil erosion, that do depend on the climate.

And while latitude may not tell you where the mountain ranges are, latitude plus knowing where the mountain ranges are tells you where the rainshadows are.

Time zones? Not really.

Nope, although that's one of the applications for which having straight-line meridians would be useful, and that mostly means using a cylindrical projection (which also has straight-line parallels) anyway.

Unless you really want to use the Craig retroazimuthal for some reason.

[Atarimaster]

99% of the source maps used to demonstrate map projections on this forum or on Daan or Atarimaster's websites are more-or-less natural-color maps

To clarify:
daan uses a modified version of derivative of NASA’s Blue Marble satellite images, so these are natural colors (probably with a bit of editing), while I’m using the Natural Earth II images, which “portrays the world environment in an idealized manner with little human influence”.

except that they like to recolor the oceans for some reason

The reason is that they have a better contrast than the original satellite images (having “almost black” oceans). In my opinion, the shapes of the continents (which are important to evaluate map projections) come out better this way.
On top of that, I simply think that the Natural Earth II images are more beautiful than satellite imagery, so that was a purely aesthetic choice.

Nope, although that's one of the applications for which having straight-line meridians would be useful

I agree, although I have seen atlases which use curved meridians for time zone maps.

and that mostly means using a cylindrical projection (which also has straight-line parallels) anyway.

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

Kind regards,
Tobias

[Milo]

I see your point, but I don't think that curving the parallels a bit is comparable with the Dymaxion.

Oh, sure, but I'm an all-or-nothing kind of guy. Keep in mind, I'm the guy who couldn't even accept compromise projections until I invented a distortion metric that shows how they're better than conformal and equal-area projections.

Anyway, while the curvature in a typical lenticular projection is less extreme, the improvement in fidelity is also likewise less extreme, and is mostly limited to some remote areas like Alaska. 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.

daan uses a modified version of derivative of NASA’s Blue Marble satellite images, so these are natural colors (probably with a bit of editing)

I find that the color Daan has in the oceans looks unnaturally pale. The color the sea has when viewed "up close" varies from photo to photo, but can get pretty dark and is pretty much never as pale as in those maps. It's a rather jarring contrast with the more realistic-seeming colors on land.

Your oceans are more reasonable in color, though they're obviously modified by bathymetry, so not just a natural color representation. Oddly, while your oceans are darker than Daan's, your land is paler, especially the greenery, which looks rather faded.

The source map I usually use is taken directly from the Blue Marble series, and so accurately shows the colors of Earth as seen from space. (Contrast actual photos from space, such as the original "Blue Marble" from Apollo 17, an earlier picture from Apollo 8, or the very earliest one from ATS-3, which does look unreasonably dark to me.)

Nope, although that's one of the applications for which having straight-line meridians would be useful

I agree, although I have seen atlases which use curved meridians for time zone maps.

I'm going to just go ahead and call that out as a really bad decision, comparable to using Mercator for density maps.

and that mostly means using a cylindrical projection (which also has straight-line parallels) anyway.

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

Well sure, that also has straight (but not parallel) meridians, but it's also relevant that time zones matter least near the poles. Inside the arctic circle, there are long periods of permanent day and permanent night, and while time-of-day is not quite completely irrelevant during this (the sun still moves around the sky a little, even if it never crosses the horizon), it's greatly reduced in importance. And as you approach the poles, the time zones get narrower. If you built a house exactly over the south pole, you could have a room where clocks need to be set to different time zones depending on what wall they're hanging on. In practice, Antarctic research bases frequently use the time zone of their home country, rather than the physical time zone of wherever their base is built (see here, here, etc.).

Given that time zones break down near both poles (north and south), and become more concrete as you approach the equator, I think that a cylindrical projection is more suitable than an azimuthal one.

Though, I suppose it helps that I never really got used to analog clocks anyway. I prefer digital ones.

[Atarimaster]

Greenland, which is seriously squashed in every flat-pole projection I know of, pseudocylindric or lenticular.

Ummm, do you really mean flat pole, or are you referring to a pole line?
Because Winkel Tripel and the A4 projection are the only flat pole lenticular projections I know – and I don’t think Greenland looks seriously squashed here; neither in lenticulars with a curved pole line, e.g. Frančula XIII, Canters W13 or any of Ginzburg’s lenticular projections.

It does get hairy in equal-area lenticular projections with a pole line. I kept an eye on Greenland when I “designed” (set in quotes because dragging sliders isn’t exactly designing) the Wagner BCW-E, but it still is very squashed. I could have improved on that by choosing a shorter pole line but I didn’t want to do that.

Your oceans are more reasonable in color, though they're obviously modified by bathymetry, so not just a natural color representation. Oddly, while your oceans are darker than Daan's, your land is paler, especially the greenery, which looks rather faded.

Actually, it’s not that odd because we were using completely different source images.
The original image I used has very pale colors all over the map, which is a very good thing if you want to put labels for cities, rivers, mountains etc. on it. But since I don’t do that on my website, I felt darker colors would be more appropriate, so I just dragged the gradation curve a bit until it looked good to my eyes. A graphic designer surely would be able to come up with a better result.

I'm going to just go ahead and call that out as a really bad decision, comparable to using Mercator for density maps.

As I’ve said, I would prefer a cylindrical projection, too – most likely the equirectangular, with standard parallels set to a value that depends on the paper sheet on which it is going to be printed.
However, I also have to say that the curved-meridian timezone maps do not look any less comprehensible to me.

Given that time zones break down near both poles (north and south), and become more concrete as you approach the equator, I think that a cylindrical projection is more suitable than an azimuthal one.

Most cylindrical timezone maps are truncated in the North and South (most likely for the reasons you mention) – you can do that on an azimuthal projections as well, see my quick & dirty example below. (This one of course looks terribly illegible but that’s due to the sloppy graphic representation, not the projection).

Though, I suppose it helps that I never really got used to analog clocks anyway. I prefer digital ones.

See, and I prefer analog clocks. Maybe that’s why I like the azimuthal timezone map.
Don’t get me wrong, in an atlas, I surely would use a cylindrical one, but for a timezone poster, I’d consider using the azimuthal variant.

[Milo]

Ummm, do you really mean flat pole, or are you referring to a pole line?

Umm, yeah. Edited to reduce confusion.

and I don’t think Greenland looks seriously squashed here

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.

Here is an orthographic projection centered on Greenland (specifically 42°W 72°N):

greenland.png (134.62 KiB) Viewed 5630 times

Note that Greenland is properly much taller than it is wide. Any projection that portrays is as merely squarish (as tall as it is wide), let alone wider, is squashing it.

Most cylindrical timezone maps are truncated in the North and South (most likely for the reasons you mention) – you can do that on an azimuthal projections as well, see my quick & dirty example below.

Sure, but azimuthal projections have lower distortion in the near hemisphere (including near the pole) and higher distortion in the far hemisphere (including near the equator), while a cylindrical distortion pushes out the worst distortion in both hemispheres to the poles, while having less distortion in the equatorial regions of the far hemisphere (which is important, because the tropics and subtropics are almost the only part of the southern hemisphere that's inhabited).

Having the time zone lines converge near the north pole also makes it harder to put in any labels there.

See, and I prefer analog clocks. Maybe that’s why I like the azimuthal timezone map.
Don’t get me wrong, in an atlas, I surely would use a cylindrical one, but for a timezone poster, I’d consider using the azimuthal variant.

Fair enough

Artistic fancy is a pretty good excuse for creating a map that has little practical merit. I've done so myself