The chord-length distribution of a polyhedron

2019/11/06 by Ciccariello, Salvino
#53C80 #60Dxx #81Uxx #FOS: Physical sciences #Mathematical Physics (math-ph)

paper · doi:10.48550/arxiv.1911.02532

Abstract

We show that the chord-length distribution function [γ"(r)] of any bounded polyhedron has an elementary algebraic form, the expression of which changes in the different subdomains of the r-range. In each of these, the γ"(r) expression only involves, as transcendental contributions, inverse trigonometric functions of argument equal to R[r, Δ₁], Δ₁ being the square root of a 2nd-degree r-polynomial and R[x,y] a rational function. Besides, as r approaches one boundary point (δ) of each r-subdomain, the derivative of γ"(r) can only show singularities of the forms (r-δ)^-n and (r-δ)^-m+1/2 with n and m appropriate positive integers. Finally, the explicit algebraic expressions of the primitives are also reported.

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