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I mean I just made up the ratio but the 2 quantities are well-known things… a fractal would end up at like 0.7 or something for eg. Legible things would be 1
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Graph shortest path is very unlikely to honor the triangle inequality. Trying to apply spatial metaphors to graphs seems ill-fated, at best you can try to have edges represent orientational metaphors a la Lakoff and Johnson.
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I don’t understand what you’re talking about An illegible environment will have wiggly paths that will on average be longer than Euclidean shortest path, You’re not applying the triangle inequality on the graph. You’re applying it in the embedding space where it will work fine
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Ie denominator will always be equal or bigger. An open plaza will give you 1. A 1E+1N path on a Manhattan grid will give you sqrt(2)/2 ~ 0.7. A complicated maze will give you very low ratios. The average ratio for all pairs of points will measure legibility decently well.
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A lot of graphs aren't planar! If your graph *is* planar, then each vertex has a direct relationship to the nearby vertices on the plane, but if it isn't, then it may only have an indirect relationship to the neighbor, per whatever layout you've chosen.
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