Interesting projection

[PeteD]

8867E9A4-B9B4-4470-B92D-704A66D42780.jpeg (99.82 KiB) Viewed 3320 times

[mapnerd2022]

Good map projection for these recent events, showing what will happen to the world if they keep letting Putin get away with all his barbarian actions.
Everything sucked into a black whole.

[Milo]

Hmm. From what I can tell, this projects the sphere onto a different non-Euclidean three-dimensional object (possibly the pseudosphere, which is interesting, since it implies I'm not the only one thinking about hyperbolic map projections), and then depicts that three-dimensional object using orthographic/perspective projection (a concept better known from the Raisz armadillo projection)?

This is rather silly from a practical perspective. Since hyperbolic space is curved the opposite way from spherical space, projecting from one to the other would result in more distortion than projecting onto flat space. But it's an interesting artistic statement. What's the context you originally encountered this in? This seems like the kind of thing where understanding the point the artist was trying to make is important.

[PeteD]

It's an album cover.

[Atarimaster]

My idea was that they had a world map (possibly Miller projection or something) and then used some graphic software that projects images to 3D objects. Pseudo graticule lines were added in the process to emphasize the “funnel” effect.

Meaning that there no knowledge of or interest in map projections was involved, it’s just a matter of artistic artwork.

[Milo]

Pseudo graticule lines

Okay, now that you mention it... I hadn't immediately noticed anything wrong with the graticule, but when I'm looking at it critically the meridian passing through South America is quite obviously wrong.

With an even closer look, I'm also now noticing that the positioning of Indonesia and Australia is totally wrong. They're drawn north of Siberia, instead of off the Malay peninsula like they should be (the Malay peninsula itself barely recognizable, but at least in about the right position of you know what to look for). I'm honestly having a hard time imagining how anyone could have ended up with that mistake, and it suggests it's more a matter of an artist cramming roughly-familiar shapes wherever there's room for them, rather than a software-automated rendering.

That definitely supports the idea that this isn't a serious projection... although I maintain that I could create a proper mathematical analogue, if I wanted to.

[mapnerd2022]

I actually was inspired by the Mollweide and the Hammer to propose 3 literal projections of an ellipse, meaning wraping an ellipse around the globe and then project it orthographically, stereographically and gnomonically. I just haven't done it yet, but I thought it would be interesting.

[Atarimaster]

it suggests it's more a matter of an artist cramming roughly-familiar shapes wherever there's room for them, rather than a software-automated rendering.

I think you are right: Yesterday, I experimented a bit with a graphics application that can project images to a fistful of predefined 3D solids. Something like that funnel (or whatever you might call it) isn’t among them, but the other solids I’ve tried suggest that the positions of the continents that is shown on the cover cannot be obtained by any rendering.

However, I came up with a set of new projections – I’ll call it my Series of Terrible Two-World Projections.
From top to bottom, the input images were Mercator, plate carrée, and Lambert equal-area cylindric:

terrible-two-world-projections.jpg (43.08 KiB) Viewed 3286 times

I think they are good for… umm, probably nothing at all.

Kind regards,
Tobias

[mapnerd2022]

The last one is like an even more distorted or worse Fournier II.

[Milo]

Er... huh? What 3D shape are you using there? A sphere, for some ironic circular logic? (Pun totally intended.)

From what I can tell, what you're doing is projecting the sphere cylindrically (with the Mercator version cut off at 85° so it has square aspect ratio), halving longitude (a la Aitoff/Hammer), projecting the resulting image back to one hemisphere of the sphere using a different cylindrical projection (cylindrical equidistant, always, so in the middle case it's actually the same projection after all, corresponding so far to the methodology of the actual Aitoff projection), and then mapping that sphere using a perspective projection?

I've managed to produce maps pretty similar to yours by performing that process using just G.Projector, so I think that's right. I'm not sure about some of the finicky details, like exactly what height your perspective projection is taken from (I'm pretty sure it's not just an orthographic projection, at least, though it's high enough that it's almost one), but my imitations still look too similar to be mere chance.

Incidentally, using a rectangular image as a texture map for a 3D-rendered sphere (which I think is what your software is doing using the equirectangular projection) is one application for which I would recommend my dihedral projection, since it has the same resolution-efficiency (this is actually one of the main applications that I invented that metric for!) as the equirectangular projection while having less distortion at the extreme points.

The last one is like an even more distorted or worse Fournier II.

...Wait, there is an actual serious projection that looks like this? Now I have to look this up.

Found it (sorta).

...Okay, from what I can tell, it's a worse version of Mollweide? It's a pseudocylindrical projection with the same envelope, but parallels are spaced farther apart near the equator and closer together near the poles, which is the opposite of what you'd want to do if you wanted to bring it closer to conformal.

For all that, though, it still doesn't look anywhere near as bad as the Atarimaster Terrible III projection.