Nitpick: you apply a transformation where they are less distorted. There is no "distorsionless" mapping from R^2->sphere. A more accurate approach would be to use spherical geometry to calculate the actual areas, but I doubt that it would change the values too much though.
Also I would be curious what was the exact projection you used and how you choose the parameters for each country.
That point is obtained by computing the midpoint of two random points from the border of the country.
I know it is not the best (if the country is not convex, then the midpoint isn't necessarily inside it), but it works. The code is at the end of the post, if you want to take a look at it.
Thanks for pointing that out though, I'm going to add this to the post.
Edit: you were right, it wasn't even the Mercator projection, it was the equirectangular projection. Thank you!
edit: They used "azimuthal projection centered in each country". That's good!
At the top would be some country with artificially-defined borders that have not since been reshaped by war or treaty. At the bottom would likely be the most "historied" country.
(Then again, at the bottom might just be Canada or Russia, since they have so much jagged coast to count. Perhaps, for the parts of a country that abut international waters instead of another country, we could use the political boundaries of the country's coastal waters surrounding that coast, rather than the boundaries of its landmass.)
http://users.math.yale.edu/~bbm3/web_pdfs/howLongIsTheCoastO...
The figure under "The Richardson Effect" on this page has some examples of how measured length of coastlines scales with the the scale of measurement:
http://www.vanderbilt.edu/AnS/psychology/cogsci/chaos/worksh...
Regular convex n-gons approach a circle in terms of shape as n increases. A square represents n=4. For any n>4, the polygon will be closer to a circle in terms of shape than a square, no?
ADDED: Right. Thank you for pulling my head out of abstract, regular convexness. IRL FTW. :-)
(In GIS, "generalization" is what you might also call "simplification" - reducing the vertex count of the borders so you have less data to deal with.)
Take Scarborough Reef (aka Scarborough Shoal) for example: #6 on the list with a Roundness of 0.9. It only has four vertices, a simple squarish quadrilateral. Is that what it is really shaped like? You be the judge:
https://commons.wikimedia.org/wiki/File:Scarborough_Shoal_La...
It's interesting, I think both of these metrics reward largely the same thing, independent of actual approximate shape, which is a lack of irregularity in their borders and many degrees of symmetry.
In practice I don't think this affects the computation much, since the overseas territories have relatively small areas.
