Stochastic Geometry of Two-Dimensional Fiber Assemblies:Part 3 : Geometrical Properties of Polygons Formed by Straight Line Segments Randomly Positioned and Oriented
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概要
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It is clarified that the orientational anisotropy affects the formation and distribution of polygons in a planar random system of line segments. The probability, P (n), of finding an n-sided polygon in an anisotropic line system is related to the same, P (n |0), corresponding to the isotropic case by the equations : P (n) = (2GI-8/π<SUP>2</SUP>-8) P (n|0), for n=3 or n≥5, andP (4) = (2GI-8/π<SUP>2</SUP>-8) P(4|0)+(π<SUP>2</SUP>-2GI/π<SUP>2</SUP>-8) where, I and G are defined, in terms of the density function, q (θ), of orientation distribution, as I=∫<SUP>π/2</SUP><SUB>-π/2</SUB>∫<SUP>π/2</SUP><SUB>-π/2</SUB>q (θ) q (θ') |sin(θ-θ') |dθdθ' and G=∫<SUP>π/2</SUP><SUB>-π/2</SUB> (∫<SUP>π/2</SUP><SUB>-π/2</SUB>q (θ) |sin (θ-θ') |dθ) <SUP>-2</SUP>dθ' The validities of these theoretical estimations are practically verified through Monte-Carlo simulations for two special cases.
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