Algorithm for density of angle distribution in random sections of polyhedron

Abstract

Scanning electron micrographs of solid materials are a source of analysis for hardness, defects, etc. A micrograph is a two-dimensional image of a material being cut. Typical photomicrographs contain about a hundred thousand structural elements, such as grain boundaries, boundary junctions, etc., which are not easy to analyze with the naked eye. Currently, computer vision can be used to analyze the geometric properties of structural elements. One of the important properties is the cross section of the grain, which is the result of cutting the material. In the article we turn to one of the geometric properties of random grain cuttings. The statistical properties of the cross sections of grains in an alloy are equivalent to the statistical properties of the cross sections of individual grains by random planes if the grains do not touch each other. We present two stages of mathematical analysis of the grain cross-sections. First, using the algorithm for generating random planes with a uniform distribution, a method is proposed for generating random sections of polyhedra. The method can be useful in stochastic geometry and computational geometry, as well as in materials science. Second, sections of polyhedra with random planes are considered, and a numerical algorithm for estimating the distribution density of angles of convex polyhedra is also implemented. Examples of distribution densities in random sections for a cube, a regular triangular prism, and a truncated prism are given. The developed algorithm and the obtained distribution densities are of practical use in the analysis of micrographs of thin sections of crystalline alloys.