Surprised! Would have expected Vatican City. Which turns out to not actually be round at all (said as a prototypical geographically-ignorant American).
Something is off with his dataset though, while №17, Monaco, looks quite round-ish in the image on the site, Monaco is actually among the more elongated and rectangular of the countries.
Not sure I like the method for computing roundness. My instinctual approach would be to look at the value 4πA/P² which has a circle=1 and the smaller values indicating less roundness (a square would be ≈0.79, a triangle ≈0.6)
I had a similar objection and idea for alternative metric (with area and perimeter). But for most countries, the perimeter is ill-defined. See: https://en.wikipedia.org/wiki/Coastline_paradox
There does not exist an adequate methodology to determine the length of a nation's coastline. Even if there did exist such a methodology, the various surveying bodies of the world's governments do not agree on any methodology. So not only are the actual lengths of a country's coastlines undefined, we don't even agree on the least bad way to calculate it.
Areas are relatively reliable despite the fact that the methodology can be wildly different between two surveying agencies. Area errors tend to cancel out. You have a little more land here, a little less land there. But perimeter error nearly always increases the perimeter, and rarely ever decreases it.
The dataset defines the countries as polygons. The perimeter of a polygon is well-defined.
Your argument that perimeter error nearly always increases the perimeter seems completely wrong to me. In a perfect snapshot of a coastline, the perimeter would be fractalized at the size of a grain of sand and would define a maximum perimeter. Any approximations would tend to smooth this out and reduce the perimeter.
But regardless of whether we can make a perfect measurement of a coastline, for the purposes of this calculation, we're looking at the country represented as a polygon on the surface of a sphere. Both the area and perimeter of such a polygon are well-defined quantities. If it makes you feel better, we can say that we're not determining the roundness of the country but the roundness of the representation of the country in the dataset (which is what the original story was also doing).
Go look at Iceland on a map. Don't just imagine it in your head, actually look at it. Now look at a map of Jordan. Ask yourself which of these countries is more circular, and then ask yourself which of these countries your metric will tell you is more circular.
And as I think about it, this would make Island archipelagos maybe not show up as such extremes since we're ignoring the separations between the islands.
> In short: the equirectangular projection distorts things, making countries closer to the poles look bigger while countries closer to the equator look smaller; the azimuthal projection accounts for this.
This isn't quite right. Equirectangular projections stretches things sideways near the poles. So a country that's very circular that is near the poles will be a wide flat oval in in the projection. Consider a circular tarp that's dropped just a few feet from the South pole. It will stretch just thousands of a degree of latitude, but will stretch a hundred or more degrees of longitude.
This is actually something that the much-maligned Mercator projection is actually good at. A circle on the globe will be a circle on the map. A circle near the poles will be a much, much bigger circle than a circle near the equator, but since you're talking about the ratios, to a significant extent the differences will cancel out.
The azimuthal projection is not a global projection. Unlike Mercator, you can't just slap the whole world into an azimuthal projection and expect to get a globally useful map. You're not going to hang a map of the entire world in an azimuthal projection in a classroom, for instance. But for a local areas, say, just one single country, an azimuthal projection will be direction preserving from the central point, which is a useful property. So what the paper does, is if you have 200 countries or whatever, it defines 200 different azimuthal projections that are only useful in and around each country. And it uses each country's own projection to calculate how circular it is.
The author declined to mention specifically which azimuthal projection they're using, which is unfortunate. Azimuthal means that direction from a central point are preserved. If this azimuthal projection is also equidistant, meaning equal distances from the central point are equal in the projection, then they're golden.
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[ 3.0 ms ] story [ 44.1 ms ] threadSurprised! Would have expected Vatican City. Which turns out to not actually be round at all (said as a prototypical geographically-ignorant American).
https://youtu.be/8mrNEVUuZdk
Something is off with his dataset though, while №17, Monaco, looks quite round-ish in the image on the site, Monaco is actually among the more elongated and rectangular of the countries.
A pentagram would have 0.26.
Areas are relatively reliable despite the fact that the methodology can be wildly different between two surveying agencies. Area errors tend to cancel out. You have a little more land here, a little less land there. But perimeter error nearly always increases the perimeter, and rarely ever decreases it.
Your argument that perimeter error nearly always increases the perimeter seems completely wrong to me. In a perfect snapshot of a coastline, the perimeter would be fractalized at the size of a grain of sand and would define a maximum perimeter. Any approximations would tend to smooth this out and reduce the perimeter.
But regardless of whether we can make a perfect measurement of a coastline, for the purposes of this calculation, we're looking at the country represented as a polygon on the surface of a sphere. Both the area and perimeter of such a polygon are well-defined quantities. If it makes you feel better, we can say that we're not determining the roundness of the country but the roundness of the representation of the country in the dataset (which is what the original story was also doing).
https://www.google.com/maps/place/Iceland
https://www.google.com/maps/place/Jordan/@31.25,34.9,7z
This isn't quite right. Equirectangular projections stretches things sideways near the poles. So a country that's very circular that is near the poles will be a wide flat oval in in the projection. Consider a circular tarp that's dropped just a few feet from the South pole. It will stretch just thousands of a degree of latitude, but will stretch a hundred or more degrees of longitude.
This is actually something that the much-maligned Mercator projection is actually good at. A circle on the globe will be a circle on the map. A circle near the poles will be a much, much bigger circle than a circle near the equator, but since you're talking about the ratios, to a significant extent the differences will cancel out.
The azimuthal projection is not a global projection. Unlike Mercator, you can't just slap the whole world into an azimuthal projection and expect to get a globally useful map. You're not going to hang a map of the entire world in an azimuthal projection in a classroom, for instance. But for a local areas, say, just one single country, an azimuthal projection will be direction preserving from the central point, which is a useful property. So what the paper does, is if you have 200 countries or whatever, it defines 200 different azimuthal projections that are only useful in and around each country. And it uses each country's own projection to calculate how circular it is.
The author declined to mention specifically which azimuthal projection they're using, which is unfortunate. Azimuthal means that direction from a central point are preserved. If this azimuthal projection is also equidistant, meaning equal distances from the central point are equal in the projection, then they're golden.