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DISTRIBUTIONS OF POINTS IN EUCLIDEAN SPACE
RANDOM LINES IN A PLANE AND IN SPACE
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applied axis centre characteristic function chosen at random circle of radius consider convex figure convex region corresponding covered Crofton's curve defined Deltheil diameters direction cosines distance distributed at random domain equal equation Euclidean space expected number fixed number formula geometrical probability given gonal Hence independently distributed infinitesimal integral interval of length invariant Jacobian joint distribution lattice points Lebesgue measure line of curvature lines intersecting mean free path mean value method nearest needle non-overlapping number of intersections number of points obtained orthogonal matrix parameter space particles perimeter perpendicular point inside Poisson distribution Poisson field Poisson variate polar coordinates prob probability density probability distribution probability measure problem projection random chord random intervals random plane random points random variable rectangle sin0 solid angle sphere square stars Suppose taken theorem theory three dimensions three-dimensional three-dimensional space tion transect transformation triangle uniformly distributed variance zero