> In short: the equirectangular projection distorts things, making countries closer to the poles look bigger while countries closer to the equator look smaller

> 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.