If you pick 3 points at random in the unit disk, the probability that they lie on a perfectly straight line is zero.
However, the probability that the curvature of the circle they lie on is less than ε is, in the limit of small ε, equal to:
1024 ε / (525 π²)
≈ 0.2 ε
![AI Overview The average curvature of the circle determined by three random points uniformly distributed in a unit disk is infinite (\(E[\kappa] = \infty\)), because the expected radius of the circumcircle is infinite. When three random points land very close to each other, the triangle's area approaches zero, making the circumradius approach infinity and the curvature approach zero. However, conversely, very large circumradius configurations do not balance out the heavy tail of near-collinear or widely spread small-area triangles in bounded domains.](https://cdn.bsky.app/img/feed_thumbnail/plain/did:plc:nyxiyurvidf33xn5bcsotkvc/bafkreidgs23nzqjecs47m2mkpkqm2gtgzecqzmkdsl7ojnodqooochmedy)
