What's a systematic algorithm for finding the best approximation (minimal under/overshoot of area) when you are given a raster or vector image representing the "real" borders? Or it just trial and error?
For a raster image, you could count the number of true and false positive pixels and true and false negative pixels. Then use statistical metrics for binary classification, like sensitivity and specificity. I guess you could even make an ROC curve by measuring the true positive rate and false positive rate for varying number of edges in the model. I guess for a vector image you could do the same thing, just using the sum of overlapping and non-overlapping areas instead of pixel counts?
Oh I was thinking about something else and should have worded my question differently: for a given number of vertices, how do you find the coordinates that cover most of the area. So for instance for 3 vertices (triangle): where do you place the three points so that you cover as close as 100% of the area as possible? Overshooting would be allowed, ie a triangle that has an area of 120% of the US would be better than one that has 70%.
One important thing to note, because I see this a lot: this doesn't mean coastlines are literally infinite in length; it's a demonstration of how our current measurement system is flawed when it comes to coastlines. Coastlines can be measured in a way that'd make them finite, however the system works well enough that (it sounds like) no one has bothered to come up with a better, widely adopted alternative.
Also, I think I've read this can be applied to borders in general? That they use the same system for measuring the borders between countries, states, cities, etc. so any border could become infinite if the precision gets high enough.
I find it interesting that (at least in my opinion) after 15 that I have to go all way to 40 before it seems like a solid improvement over what 15 offers.