Post #3179039
2025-09-02 14:51 UTC
@gregeganSF@mathstodon.xyz area nearest 4 is largest, then area nearest 3. Statistical likelihood of picking points in these two are higher than picking one in area nearest two.
Not sure about extrapolating from there to likelihood of the line’s directions. I know one ray has to cross 3 or 4 for each pair simply because any line will cross two edges.
Hmm, from one point taken randomly, for any second point, the odds of crossing an edge are going to be proportional to the number of points in that direction, which again weighs toward the higher areas near 3 and 4. More points to be selected at random means higher odds of selection .
That third paragraph makes me think an algorithm picking pairs and determining their crossings would favor 3 and 4. Numeric models approach algebraic at large n, so 3 & 4.
Replies (1)
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@bojidar_bg@mastodon.social 2025-09-02 15:08
@cascheranno@hachyderm.io @gregeganSF@mathstodon.xyz Hmm.. but by the same token, if two points "near" 4 get picked together, wouldn't you get a line roughly parallel to 4, thus crossing 2 and 3? I have an odd feeling you end up getting a nasty cancellation that makes any pair equally likely 🤔 (edit: actually- I have an almost-proof using barycentric coordinates and the way they interact with ratios of areas and sampled points that shows all pairs of sides are equally likely, but don't trust me before I've written it out)