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

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