The chord-length distribution of a polyhedron

TL;DR

This paper derives an explicit algebraic form for the chord-length distribution function of any bounded polyhedron, revealing its structure across different subdomains and describing singularities at boundary points.

Contribution

It provides a novel elementary algebraic expression for the chord-length distribution of polyhedra, including boundary behavior and primitive functions.

Findings

Explicit algebraic formulas for $ ext{γ''(r)}$ in different subdomains.

Descriptions of singularities at boundary points.

Explicit primitives of the distribution function.

Abstract

We show that the chord-length distribution function $[\gamma"(r)]$ 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 $\gamma"(r)$ expression only involves, as transcendental contributions, inverse trigonometric functions of argument equal to $R[r,\,\Delta_1]$, \,$\Delta_1$ being the square root of a 2nd-degree $r$-polynomial and $R[x,y]$ a rational function. Besides, as $r$ approaches one boundary point ($\delta$) of each $r$-subdomain, the derivative of $\gamma"(r)$ can only show singularities of the forms $(r-\delta)^{-n}$ and $(r-\delta)^{-m+1/2}$ with $n$ and $m$ appropriate positive integers. Finally, the explicit algebraic expressions of the primitives are also reported.