Experimental projections

[Atarimaster]

… and, as a side note:

with a little extension for Greenland’s coast

That extension was made using a graphics editor.
I just realised that things like that can be done in Geocart using the “Custom…” setting in the Boundaries dialogue. Moreover, you can use almost the same XML code to avoid the repetition of Greenland’s coast in the Eastern part if you prefer it like that.

Of course at some point I had inspected the custom setting, but I never had any use for it so I forgot it.
Very nice feature, thank you!

[brsr]

Hi, I'm new here.

I've been working on a variation of the Chamberlin Trimetric projection. For the most part it just looks like the Chamberlin Trimetric, but when the control points are pushed out to an entire hemisphere, it maps the hemisphere to a triangle. Maybe too much distortion to be practically useful, but it would make a good logo for something.

hemisphere.png (26.25 KiB) Viewed 4225 times

It looks much better on the use cases that the Chamberlin Trimetric is useful for. If anyone is interested, I have a writeup on the new projection on GitHub.

[PeteD]

1. As I’ve said, I don’t think I’ve ever seen this specific arrangement but of course that doesn’t mean that it doesn’t exist. It actually seems a bit unlikely to me that no one ever tried this. So, have you ever seen this somewhere?

I've never seen it published anywhere, but it does seem fairly obvious, and it's occurred to me before, so there's a good chance that there's a map like this published somewhere.

The problem with it is that you still have to cut through either Greenland or Iceland -- or make an extension for one of them as Tobias did. To solve this problem, you can interrupt the Atlantic along a line that's not a meridian:

unequal_hemispheres.png (90.1 KiB) Viewed 4216 times

Please excuse the line connecting the two halves of Antarctica.

And while from a perspective of reducing distortion, there's no point in splitting a cylindrical projection into two hemispheres, you can certainly do it for fun. If you interrupt both the Pacific and the Atlantic along lines that aren't meridians and position the hemispheres on top of each other rather than beside each other, then you can get what I call the beer glass aspect of the equirectangular projection:

beer_glass.png (89.26 KiB) Viewed 4216 times

In this case, the curves along which the projection is interrupted are defined such that if you rolled the map up to connect the two North Poles, you'd get half a torus with the reduced western hemisphere on the inside and the enlarged eastern hemisphere on the outside. Of course, the paper wouldn't actually bend to make a half a torus without more gores, but the length of each parallel in each hemisphere is correct for rolling the map into half a torus.

2. How would you name a configuration like this?

Sorry, I can't help you there.

[PeteD]

In this case, the curves along which the projection is interrupted are defined such that if you rolled the map up to connect the two North Poles, you'd get half a torus with the reduced western hemisphere on the inside and the enlarged eastern hemisphere on the outside. Of course, the paper wouldn't actually bend to make a half a torus without more gores, but the length of each parallel in each hemisphere is correct for rolling the map into half a torus.

Correction: it was originally half a torus, but then I compressed the projection horizontally, both to move the standard parallels away from the equator and to make it look more like a beer glass, so now it would make 151.2° of a torus.

[Atarimaster]

it does seem fairly obvious (…) so there's a good chance that there's a map like this published somewhere.

My thoughts exactly.

The problem with it is that you still have to cut through either Greenland or Iceland -- or make an extension for one of them as Tobias did. To solve this problem, you can interrupt the Atlantic along a line that's not a meridian

Good idea!
Still have to decide wether I prefer the asymmetry of your variant or the little “bump” in my current version, made with the custom boundaries mentioned above, which still could use a bit of refinement:

unequal-hemispheres-with-custom-boundaries.jpg (72.34 KiB) Viewed 4215 times

If you interrupt both the Pacific and the Atlantic along lines that aren't meridians and position the hemispheres on top of each other rather than beside each other, then you can get what I call the beer glass aspect of the equirectangular projection

Oh, I like it!
If there was a beer glass with a map printed on it this way – I’d buy it!

Kind regards,
Tobias

[PeteD]

Still have to decide whether I prefer the asymmetry of your variant or the little “bump” in my current version, made with the custom boundaries mentioned above, which still could use a bit of refinement:

Yes, now that I look at it again, the asymmetry does make it look a bit weird, but this is very easy to fix:

unequal_hemispheres_azimuthal_equidistant.png (89.63 KiB) Viewed 4209 times

I just took the boundaries from the beer glass map.

As you can see, getting it not to plot the annoying lines connecting the two halves of Antarctica isn't quite so easy to fix

[PeteD]

Another drawback to these "unequal hemisphere" maps has occurred to me, at least for azimuthal projections. As we know, the azimuthal equal-area, azimuthal equidistant and stereographic projections all portray one hemisphere fairly accurately, but distortion increases rapidly beyond that one hemisphere. Since the right "hemisphere" is actually wider than one hemisphere, distortion becomes significant at the edges, as can be seen from the shape of Africa.

This can be solved with a little bit of Umbeziffern. The following projection is derived from the azimuthal equidistant projection via Umbeziffern in the horizontal direction only. The limiting meridian is chosen such that the width of the right "hemisphere" corresponds to 180° of the parent azimuthal equidistant projection. More specifically, the width of the right "hemisphere" is 226.8°, so the limiting meridian is (180^2)/226.8 = 142.9°:

unequal_hemispheres_azimuthal_equidistant_umbeziffert.png (88.28 KiB) Viewed 4195 times

Africa is definitely improved.

You can also apply the Umbeziffern transformation in both directions if you like:

unequal_hemispheres_azimuthal_equidistant_2D_umbeziffert.png (98.63 KiB) Viewed 4195 times

However, this Umbeziffern isn't actually necessary for the reduced left "hemisphere", which the azimuthal equidistant already portrays fairly accurately, so you could also apply the Umbeziffern transformation to the right "hemisphere" only:

unequal_hemispheres_azimuthal_equidistant_mixed.png (88.72 KiB) Viewed 4195 times

I really like these "unequal hemisphere" maps. They have fairly low distortion for world maps without splitting any landmasses other than Antarctica while keeping the traditional layout with the North Pole at the top, the South Pole at the bottom and the equator as a horizontal line in the middle.

[Atarimaster]

Nice idea with the boundary cut and the Umbeziffern!

Regarding the interior of the map, I like your last example best although I’m not so sure that the left “hemisphere” has a rounded pole and the right a spiky one. Strangely, it didn’t disturb me that my own first example has different kinds of poles, too.

Meanwhile, I’ve experimented a bit with non-azimuthal maps.
Here are some examples with distortion visualization. Red = angular deformation, green = areal inflation (in relation to value at the map’s center, not the nominal scale).

First row: Wagner VII on the left (with a bit of area-preserving horizontal stretch) and Strebe 1995.
Second row: Laskowski’s Tri-optimal.
Third row: Wagner VIII, but with an areal inflation by 1.35 at 60° N/S (instead of 1.2 as in the original).

unequal-hemispheres-demos.png (261.26 KiB) Viewed 4185 times

I’m still trying to find an example of a non-azimuthal pointed-pole projection that I like.

Kind regards,
Tobias

[PeteD]

Nice work, Tobias!

I’m still trying to find an example of a non-azimuthal pointed-pole projection that I like.

In general or in a two-hemisphere configuration? In general, I really like Györffy D-F and danseiji N and II (or danseiji I if you need a pointed-pole projection that's (quasi-)equal-area). However, in a two-hemisphere configuration, none of them come close to the azimuthal projections.

[Atarimaster]

In general or in a two-hemisphere configuration?

In a two-hemisphere configuration.
I’ve found some which I think are not too bad: Two equal-area projections, Strebe-Mollweide and Hufnagel 10 (the latter again with a slight horizontal stretch), and a compromise projection, Apian II. Both Strebe-Mollweide and Hufnagel are customizable so it’s possible that there are configurations which work better in the “unequal hemispheres” representation.