Geometric probability question.
What is the probability that the shortest paths between three random
points on a projective plane form a contractible loop?
Points on
a sphere. Paul Bourke describes a simple random-start hill-climbing
heuristic for spreading points evenly on a sphere, with pretty pictures
and C source.
Random spherical arc crossings.
Bill Taylor and Tal Kubo prove that if one takes two random geodesics
on the sphere, the probability that they cross is 1/8.
This seems closely related a famous problem on the probability
of choosing a convex quadrilateral from a planar distribution.
The minimum (over all possible distributions) of this probability
also turns out to solve a seemingly unrelated combinatorial
geometry problem, on the minimum
number of crossings possible in a drawing of the complete graph with
straight-line edges:
see also "The
rectilinear crossing number of a complete graph
and Sylvester's four point problem of geometric probability",
E. Scheinerman and H. Wilf, Amer. Math. Monthly 101 (1994) 939-943,
rectilinear
crossing constant, S. Finch, MathSoft, and
Calluna's pit,
Douglas Reay.
Random polygons.
Tim Lambert summarizes responses to a request for
a good random distribution on the n-vertex simple polygons.
