All angles. See the graph above in the original tweet. It’s 3% difference even in the worst case.
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Replying to @pcwalton @Vector_GL
This is from a Monte Carlo simulation of randomly placed edges and calculating exact areas over a pixel square.
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Replying to @pcwalton @Vector_GL
Tentative conclusion: Not worth calculating exact trapezoidal area coverage. Better to just use a polynomial approximation based on distance
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Replying to @pcwalton
Interesting. My simple geometrical analysis shows a huge difference between linear and diagonal.pic.twitter.com/yjJaIszjDO
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Replying to @Vector_GL
I’m confused. Your pixel square doesn’t have length 1 in both drawings (in the second it’s sqrt(2)/2).
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Replying to @pcwalton @Vector_GL
The area of your second square is 0.5, so if I’m not mistaken the fraction of the total area is 0.25, same as the first square.
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Replying to @pcwalton
I’m using proportions. Count the triangles. Each quadrant has 4. Each is 1/16 of the total. Only 2 are covered, i.e. 1/8 of the total pixel area.
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Replying to @Vector_GL
But your pixel square doesn’t have side length 1, so it’s not comparable. I’m measuring exact distance from center of pixel square to edge.
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Replying to @pcwalton @Vector_GL
(Also NB: By 3% I mean off by at most 3% of the total area of the square.)
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Replying to @pcwalton
Thanks. I've done a more accurate drawing which more closely matches your results. Hope it makes sense.pic.twitter.com/nEHvplewPe
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BTW, the polynomial I came up with is -0.83570 * d^3 + 1.14191 * d + 0.5. Just did cubic regression on the Monte Carlo simulation results.
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Replying to @pcwalton
Nice! http://desmos.com plot here: https://www.desmos.com/calculator/kxkpeq8we6 …
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