The Liquid Earth projection

[saga]

This is my first post here, so I'll post what I've made recently.

This is the Liquid Earth projection, an (approximately) equal-area projection made using meshes. It came out of my research into using meshes to generate cartograms. This is similar to Justin Kunimune's Danseiji and Elastic projections, but unlike those, it uses meshes to map the sphere to itself, then maps the sphere to the plane using the Equal Earth projection. That's how it achieves a regular outer edge despite the graticule being irregular.

I also made one that I'm calling Solid Earth, which is one that expands land and compresses ocean, similar to Danseiji V and Elastic III. Compared to those two projections, this is definitely the most efficient way of fitting the Earth's land into a rectangle so far. At this point, the bottleneck is not area, but length, as large continental areas struggle to fit within the boundaries. Though this map is obviously not equal-area, the area scale is fairly consistent across non-Antarctica land. I have a full writeup of these projections here.

[daan]

Welcome to the forums, saga. That’s an interesting inaugural posting. It looks like you’ve put a lot of work into this.

I like the Liquid Earth; I could see it being used for maps whose spatial relationships among continents is unimportant. It does a good job of preserving continental shapes on a (nearly) equal-area map. It definitely needs a finely spaced graticule, doesn’t it! I’m not so bothered that it’s not precisely equal-area insofar as the angular deformation is bounded. Do you have metrics for that?

I see some exploratory value in the Solid Earth, but it’s hard for me to imagine using it except for something like a board game. It seems as if you could achieve something similar by taking very low distortion caricatures of each of the landmasses and arranging them ad hoc by sliding them around with a mouse pointer in a graphics program, given that the map doesn’t seem to preserve much beyond continental shapes.

Cheers,
— daan

[PeteD]

Welcome to the forum!

I like your idea of a globe-to-globe mapping that anticipates the distortion caused by the globe-to-plane mapping!

As you say, distortion of the landmasses is very low at the expense of distortion of the oceans, meaning the positional relations of the landmasses are messed up. In other words, the map is very accurate when looking at one landmass at a time but not really suitable for getting an idea of how that landmass sits in relation to the rest of the world, which is absolutely fine, but then why is it important for the boundary to be regular if you can only really use the map to look at one landmass at a time?

[saga]

@PeteD Thanks! I think the regular boundary is aesthetically important, but it has practical value as well. Maps are usually displayed in a rectangle, either on a screen or on a wall or something else. If you look at e.g. Martin 2, which preserves land in a similar way, a lot of space is wasted when fitting this in a rectangle.

[Milo]

That... looks better than anything with such a ridiculous graticule has any right to. Though with the Atlantic distorted that badly, I wonder what the point of even drawing it contiguously is.

The second projection, despite supposedly being all about the continents, distorts Antarctica more than the first. Still, I doubt the inhabitants are going to complain.

If you look at e.g. Martin 2,

I think it's a shame that one draws an interruption through Antarctica when it comes so close to avoiding the need for it.

I also like the Elastic II projection later on that page. It highlights the relationship between the three major oceans (Pacific, Atlantic, Indian) in a way that I don't think I've ever seen before. Takes some concentration to identify the continents, though!

[saga]

@daan Thanks for accepting me! Here are metrics for the shape distortion on land: 50th %ile 1.096, 90th %ile 1.227, 99th %ile 1.496, 99.9th %ile 1.935, max 23.397. This means, for example, that the median point on land turns a circle into an ellipse with dimensions 1.096:1. So there isn't a good bound per se, but the distortion is low across the board. I didn't turn these into angle distortion values because I don't know the typical definition of angle distortion. Is it the maximum difference between an angle on the globe and its image on the plane?

It seems as if you could achieve something similar by taking very low distortion caricatures of each of the landmasses and arranging them ad hoc by sliding them around with a mouse pointer in a graphics program, given that the map doesn’t seem to preserve much beyond continental shapes.

I think this would be a ton of work, especially if you wanted to get islands right. Solid Earth represents islands relatively undistorted and at the same scale as the continents, which would be hard to achieve manually. Even if you only cared about the continents, you'd need to put it together manually for every map you wanted to make, instead of just being able to select the projection in a cartography program. Justin Kunimune already has Liquid Earth available in his apps (Solid Earth coming soon), and I'm looking into implementing the projections for d3. I'm not sure what software is typically used for this, but I think a clear use case for Solid Earth is for choropleth maps made to be shown at a small size.

[saga]

Though with the Atlantic distorted that badly, I wonder what the point of even drawing it contiguously is.

The second projection, despite supposedly being all about the continents, distorts Antarctica more than the first. Still, I doubt the inhabitants are going to complain.

You could definitely argue that the Atlantic distortion is an interruption in disguise. Keeping it together at least saves space though, since a real interruption introduces whitespace to the middle of the map.

Unfortunately, there just isn't enough room in Solid Earth's layout to fit Antarctica properly while accommodating the other continents. I figured that for use cases where you want to draw continents large, you probably don't care about Antarctica, so I weighted it less in the optimization. Liquid Earth is able to represent Antarctica faithfully because there's enough room for it.

[daan]

I didn't turn these into angle distortion values because I don't know the typical definition of angle distortion.

For angular deformation 𝜔, it is common to use
𝜔 = 2 arcsin(𝑏′/𝑎′)
where
𝑎′ = √(h² + 𝑘² + 2h𝑘 sin 𝜃′)
𝑏′ = √(h² + 𝑘² − 2h𝑘 sin 𝜃′)
h = √((∂𝑥/∂𝜑)² + (∂𝑦/∂𝜑)²)
𝑘 = sec 𝜑 √((∂𝑥/∂𝜆)² + (∂𝑦/∂𝜆)²)
sin 𝜃′ = [(∂𝑦/∂𝜑)(∂𝑥/∂𝜆) − (∂𝑥/∂𝜑)(∂𝑦/∂𝜆)] / (h𝑘 cos 𝜑)

What you gave is sufficient for computing 𝜔, so it’s not much of a problem.

What I intended to enquire about was the (in/de–)flation characteristics of your “nearly” equal-area projection. It appears to be tightly bounded but would be good to know more precisely. (I inadvertently asked about angular deformation instead, but am glad to have that as well.)

— daan

[saga]

What I intended to enquire about was the (in/de–)flation characteristics of your “nearly” equal-area projection. It appears to be tightly bounded but would be good to know more precisely. (I inadvertently asked about angular deformation instead, but am glad to have that as well.)

Oh, I see. I have a table for that in the longer description linked in the OP.

[saga]

For angular deformation 𝜔, it is common to use
𝜔 = 2 arcsin(𝑏′/𝑎′)
where
𝑎′ = √(h² + 𝑘² + 2h𝑘 sin 𝜃′)
𝑏′ = √(h² + 𝑘² − 2h𝑘 sin 𝜃′)
h = √((∂𝑥/∂𝜑)² + (∂𝑦/∂𝜑)²)
𝑘 = sec 𝜑 √((∂𝑥/∂𝜆)² + (∂𝑦/∂𝜆)²)
sin 𝜃′ = [(∂𝑦/∂𝜑)(∂𝑥/∂𝜆) − (∂𝑥/∂𝜑)(∂𝑦/∂𝜆)] / (h𝑘 cos 𝜑)

I did some working out, and this is exactly the definition I guessed, i.e., the maximum difference between an angle on the globe and its projected counterpart.

The way the calculation is written here, h² + k² is the squared Frobenius norm of the Jacobian, and hk sin 𝜃′ is its determinant. With that in mind, a simpler way to write this calculation would be

f = ‖J‖_F
d = det J
𝜔 = 2 arcsin(sqrt((f² - 2d) / (f² + 2d))),

where J is the Jacobian matrix.