Total Chord Length When a Random Disc Intersects Equispaced Parallel Lines

Construct the lines x = k $x = k$ x equals k for integers k k k . Centre a disc of radius r r r on ( S , 0 ) $(S,0)$ left parenthesis upper S comma 0 right parenthesis , where S S upper S has a uniform distribution on [ 0 , 1 ] $[0,1]$ left bracket 0 comma 1 right bracket . Let L L upper L be the total chord length from the intersection of the disc with the lines. This article obtains the cumulative distribution function for L L upper L .