Abstract
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.
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References
ARMITAGE, P. (1949). An overlap problem arising in particle counting. Biometrika 36, 257–266.
BOREL, E. (1925). Principes et formules classiques du Calcul des Probabilités. Traité du Calcul des Probabilités et ses Applications, Gauthier-Villars, Paris.
GARWOOD, F. (1947). The variance of the overlap of geometrical figures with reference to a bombing problem. Biometrika 34, 1–17.
GHOSH, B. (1943a). On the distribution of random distances in a rectangle. Science and Culture 8, 388.
GHOSH, B. (1943b). On random distances between two rectangles. Science and Culture 8, 464.
GHOSH, B. (1949). Topographic variation in statistical fields. Calcutta Statist. Assoc. Bull. 2 (5), 11–28.
GHOSH, B. (1951). Random distances within a rectangle and between two rectangles. Bull. Calcutta Math. Soc. 43, 17–24.
KENDALL, M.G., and MORAN, P.A.P. (1963). Geometrical Probability, Griffin, London.
MATERN, B. (1947). Metoder att uppskatta noggraunheten vid linjeoch provytetaxering. (Methods of estimating the accuracy of line and sample splot surveys.) [In Swedish. English summary.] Meddelanden fran Statens skogsforskningsinstitut, 36 (1), 1–138.
RUBEN, H. (1964). Generalized concentration fluctuations under diffusion equilibrium. J. Appl. Prob. 1, 47–68.
RUBEN, H. (1967). An intrinsic formula for volume. J. Reine Angew. Math. 226, 116–119.
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.
WHITNEY, H. (1957). Geometric Integration Theory, Princeton Univ. Press, Princeton, N.J.
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Ruben, H. (1978). On the Distance Between Points in Polygons. In: Miles, R.E., Serra, J. (eds) Geometrical Probability and Biological Structures: Buffon’s 200th Anniversary. Lecture Notes in Biomathematics, vol 23. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-93089-8_5
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DOI: https://doi.org/10.1007/978-3-642-93089-8_5
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