Greg Egan on Nostr: If you pick 3 points at random on a unit sphere, the average radius, R, of the circle ...

If you pick 3 points at random on a unit sphere, the average radius, R, of the circle that contains them is 7/6.

For 3 points on a flat disk of radius ρ, mean(R) diverges. However, the curvature, κ = 1/R, has a finite mean which can be computed exactly:

16 (3 – 16/π^2) / (15ρ)

Although it’s easy to write down the radius of the circle through any 3 points, given their coordinates, integrating the reciprocal of this over all triples of points inside a disk is unwieldy.

It’s easier to switch to a coordinate system where R is one of the coordinates, along with the location of the centre of the circle and the positions of the 3 points around the circle.

Details at https://www.gregegan.net/SCIENCE/ThreePointCurvature/ThreePointCurvature.html