
Re: Random Triangle Problem
Posted:
Aug 1, 1997 1:59 PM


Chris Thompson (cet1@cus.cam.ac.uk) wrote: : Can one extend this to show that the distribution of triangle shapes : (not just the probability of obtuseness) is the same for three points : picked independantly from the circular normal distribution as for three : lines with independant uniformly distributed orientations [Bill Taylor's : interpretation of the original problem]?
The distribution of shapes is different. For small epsilon, the probability that the directions of the lines are all within epsilon of each other is roughly proportional to epsilon with the normal distribution, but to epsilon^2 with Bill Taylor's distribution; this isn't hard to see.
To put it another way, given that the angle at A is very small, with the normal distribution it's probable that one of the other angles will also be small, but B and C won't be especially close; with the line directions independent it's probable that B and C will be close (compared with their distance from A), but neither of the other angles will be especially small.
 John Rickard

