Some Properties of Order Distance for Random Point Distributions

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distributions of various point-like objects in order to evaluate the hypothesis that an observed distribution corresponds to a theoretical distribution of points. The random point distribution has a central position in this type of study because, on one hand, a random point distribution is a uniform distribution of points and, on the other hand, many collections of objects studied by geographers may be treated as points uniformly distributed within homogeneous regions. Theoretical random point distributions may be defined for points uniformly located within the Euclidean plane or within polygons of various shapes, and properties of these point distributions have been studied extensively. The relations that exist between formulations for regions with infinite and finite area, however, have not been identified. This study compares properties of spacing measures for random point distributions defined in several ways. In section I a limit process is identified that relates the probability density functions for order neighbor distances between points randomly located in the plane and in the square with unit area. In sections II and III point distributions are constructed that have properties approximating those of theoretical random point distributions. These artificial random point distributions simplify the analysis of distance between points and are particularly useful for estimation of distance between points randomly located in regions having irregular shapes for which analytic methods are not suited for the derivation of distance relations.