Wednesday, 28 September 2016

calculus - Probability of two fixed-length line segments intersecting within a circular domain

Imagine placing a line segment P of length a on the XY plane such that its middle is at the origin, but its orientation is random (i.e. random angle). Then suppose you placed another line segment Q of length b (which is less than or equal to a) such that its centre was randomly chosen within a radius (a+b)/2 from the origin, but its orientation was also random. What would the probability of these two line segments intersecting be?



I suspect there may be an integral over the circular surface to be done here, but am not quite sure what to do! Any help much appreciated. Thank you.



This image shows you what I mean. Here the two line segments don't intersect.

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