consider two points $a$ and $b$ having integer coordinates inside two circles each of radius $r$ centred at $C_1$ and $C_2$ respectively in two dimensional Euclidean space. There are few line segments present in the environment also, whose endpoints are known. I need to calculate the probability of the line segment joining $a$ and $b$ not intersecting any of those line segments.
my approach
My solution is based on simple counting and thus brute force. I have considered every pair of points between the two circles and checked whether there is an intersection or not. I just count the number of non intersecting point pairs and divided it by number of all possible pairs.
my need
Clearly the above approach becomes infeasible when $r$ is large. Is there any better way to compute the probability? Can we obtain some closed form expression?
