> We cannot use those (xi,yi)(xi,yi) directly, because they come from a Mercator projection of the earth, thus they are distorted. I had to apply a transformation to them in order to get coordinates that aren’t distorted. In order to do that I used the pyproj library.<p>Nitpick: you apply a transformation where they are <i>less</i> 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.<p>Also I would be curious what was the exact projection you used and how you choose the parameters for each country.
Neither this, nor the post it's inspired by, is as interesting to me as a question with a much simpler-to-calculate answer: what's the <i>smoothest</i> (i.e. lowest fractal-dimension, smallest surface-area-to-volume ratio) country?<p>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.<p>(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 <i>coastal waters</i> surrounding that coast, rather than the boundaries of its landmass.)
For some reason Egypt has shown up as the most rectangular and now one of the most round countries. Goes back to how the average person's definition of these terms isn't capture by the metric. When I say a country is "rectangular" I'm thinking about straight lines and sharp corners. "Round" I guess should be the absence of corners?
Watch out for over-generalization, especially when the thing you're generalizing is a geographic shape.<p>(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.)<p>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:<p><a href="https://commons.wikimedia.org/wiki/File:Scarborough_Shoal_Landsat.jpg" rel="nofollow">https://commons.wikimedia.org/wiki/File:Scarborough_Shoal_La...</a>
I don't recognize France on spot 156. As for United States on spot 121, Netherlands on spot 93, Denmark on spot 97 etc. it looks like they have included all overseas territories. But it does look weird.
So, Egypt is both round (20) and rectangular (1)?<p>Compare:
<a href="http://pappubahry.com/misc/rectangles/" rel="nofollow">http://pappubahry.com/misc/rectangles/</a>
LaTeX nitpick: use \text for function names, like \text{roundness}, or else you have <i>roundness</i> which means <i>r</i> times <i>o</i> times <i>u</i>...
Nauru is the 2nd roundest (0.923) and the 10th most rectangular (0.917). <a href="http://pappubahry.com/misc/rectangles/" rel="nofollow">http://pappubahry.com/misc/rectangles/</a>