Derivation of the Generalized, Average Euclidean Distance Function for the PDI Model

This report derives distance functions that form the basis for the Population Density Index PDI model, which is a three-parameter square-root model for measuring discrete spatial density in finite populations. The PDI and its methods have been applied to facilities layout methodologies in submarine environments at the Naval Undersea Warfare Center Division, Newport, RI, resulting in several U.S. patent applications. The emphasis here is on the micro-population model in which the linear units are feet. The derivations relate Cartesian rectangular coordinate systems to uniform unit and nonunit lattices, as well as to the nonlattice distribution. Other proofs relate to the bounds of the calculated density measure and the density rate index called effective distance. Alternative distance functions are discussed, and examples of the numerical calculations are provided. Also derived is the algorithm for selecting a rectangular lattice conformal to a quadrilateral area and for calculating interpoint distance in a PDI lattice. A table of computer-generated unit lattice average Euclidean distances for up to 10,000 density points is included.