Author here: when calculating this I _did_ assume a uniform (area) distribution on the unit disk.

Now it does say

> Three points are chosen independently and uniformly at random from the interior of a unit circle.

which sounded OK to me at the time but I understand there could have been some ambiguity. Especially around the "uniform on area" part.

Also, I think that with rejection sampling you could get the same with 1) [0], 2) would work (provided correct scaling) [1]. No idea about 3) or the normal distribution thing you mentioned - I figured the problem was hairy enough already!

[0] https://blog.szczepan.org/blog/monte-carlo/#sampling-uniform... [1] https://blog.szczepan.org/blog/monte-carlo/

Totally agreed! Conceptually I think of the radian/length distribution as having uniformly increasing density closer to the origin - you could imagine a whole bunch of discs concentrically stacked ending in the circumference of the circle - each of them - if a constant "radius length" - will have the same "number" of points but spread out over a larger total area.

It's been a lonnnng time since my geometry university courses, but my vague memory is there are some tricky differential geometry historical problems that founder on this precise imprecision.

Fun site, thank you for the write up. I skimmed each and every matrix and assumed you did a great job.