On the Distance Between Points in Polygons
The need to evaluate the probability distribution, or at least the first few moments, of the distance between two independent uniform points, each in a planar and bounded polygonal domain, appears to surface periodically in numerous diverse areas of investigation. For some early isolated results, with applications in bombing, particle counting, and forestry sample surveys [or, more generally, problems of topographic variation -- see Matérn (1947) and Ghosh (1949)] etc., refer tb Borel (1925), Ghosh (1943a,b; 1951), Matérn (1947) and Armitage (1949). Evaluation by direct integration, or by some such device as Crofton’s ‘second’ theorem [more precisely, a slight extension of this theorem -- see Kendall and Moran (1963), pp.65-66] is at best intolerably tedious and time-consuming1, and at worst intractable. In this paper, I present a method which should enable future investigators to determine systematically, and with relatively little effort, the probability distribution and moments of the distance, once the coordinates of the vertices of the polygonal domains have been specified.
Unable to display preview. Download preview PDF.
- BOREL, E. (1925). Principes et formules classiques du Calcul des Probabilités. Traité du Calcul des Probabilités et ses Applications, Gauthier-Villars, Paris.Google Scholar
- GHOSH, B. (1949). Topographic variation in statistical fields. Calcutta Statist. Assoc. Bull. 2 (5), 11–28.Google Scholar
- GHOSH, B. (1951). Random distances within a rectangle and between two rectangles. Bull. Calcutta Math. Soc. 43, 17–24.Google Scholar
- RUBEN, H. (1964). Generalized concentration fluctuations under diffusion equilibrium. J. Appl. Prob. 1, 47–68.Google Scholar
- RUBEN, H. (1967). An intrinsic formula for volume. J. Reine Angew. Math. 226, 116–119.Google Scholar
- RUBEN, H. (1970). On a class of double space integrals with applications in mensuration, statistical physics, and geometrical probability, Proc. 12th Biennial Seminar of Canadian Mathematical Congress, 209–230.Google Scholar
- WHITNEY, H. (1957). Geometric Integration Theory, Princeton Univ. Press, Princeton, N.J.Google Scholar