Algebraic approximations of a polyhedron correlation function stemming from its chord-length distribution

An algebraic approximation, of order $K$, a polyhedron correlation function (CF) can be obtained from $\gamma\pp(r)$, its chord-length distribution (CLD), considering first, within the subinterval $[D_{i-1},\, D_i]$ full range distances, polynomial in two variables $(r-D_{i-1})^{1/2}$ and $(D_{i}-r)^{1/2}$ such that expansions around $r=D_{i-1}$ $r=D_i$ simultaneously coincide with left right $\gamma\pp(r)$ $D_{i-1}$ $D_i$ up to terms $O\big(r-D_{i-1}\big)^{K/2}$ $O\big(D_i-r\big)^{K/2}$, respectively. Then, for each $i$, one integrates twice determines integration constants matching resulting integrals at common end points. The 3D Fourier transform CF approximation correctly reproduces, large $q$s, asymptotic behaviour exact form factor term $O(q^{-(K/2+4)})$. For illustration, procedure is applied cube, tetrahedron octahedron.