Some more average distance results (Q2204290)

Summary: The purpose of this study is to present a taxonomy of expected distance functions (EDFs). An EDF is the expected (where expected is used in the strict probabilistic sense) distance formula for a given metric between two algebraically defined regions (e.g., the expected Euclidean distance between a semi-circle of radius \(r\) centred at \((x_{1}, y_{1})\) that has a known bivariate probability density function in \(r\) and \(\theta\) and a line segment beginning at \((x_{2}, y_{2})\) and ending at \((x_{3}, y_{3})\) that has a known probability density function along its length). A modest library of EDFs for various metrics (rectilinear, Euclidean, Tchebychev, etc.) between pairs of common geometric shapes (e.g., lines, semi-circles, rectangles, etc.) is presented. The taxonomy of EDFs contained herein is by no means meant to be an exhaustive list. Indeed, it is limited in scope to those considered to be of practical importance to geographic information, transportation science, and mathematical modelling professionals.