On Scatteration Measure of Objects in Space

The study on distribution of objects in space could be divided into two parts, that of form and process. Although the form of distribution could be thought including distribution pattern and spacial dispersion, neither the difference between them is recognized clearly, nor much attention has been paid to the study about the spatial dispersion. This paper mainly aims to built a index measuring how much densely or dispersively do objects distribute in space. The objects dealt with here are treated as points, lines, surfaces and solids, being called as point-objects, line-objects, surface-objects and solid-objects respectively (fig. 1), and meanwhile are treated as the ones spread in one, two and three-dimensional space. Scatteration measure of objects in one or two-dimensional space often used up to now is standard deviation or something like it. For example socalled ‘standard distance’ defined by Bachi in 1957 is calculated as follow. Sd=(1/NΣNi=1dci2)1/2 (*) where, dci, represents the distance between object i(i=1, 2, …, N) and their mean center, C. It has been introduced that the standard distance could be used to distinguish the difference of scatteration between different groups of objects. In fact, that is not correct. If we consider the square lattice points as shown in fig. 2 in the paper, the value of standard distance of the square lattice points should remain same despite of their size, since that kind of lattice points spread uniformly in a plain. But the value, Sd, of the square lattice points calculated by the way mentioned in formula (*) is not so but becomes bigger as the number of objects studied, N, increases (see formula 4 in the paper). Based on some discussing made in appendix about so-called ‘uniform point set’, the paper suggests a new measure of the scatteration of point-objects as follow: Pcb, m=GΣNi=1ΣNj=1dijb/N(2m+b)/m (**) where, Pcb, m, named as ‘scatteration measure’, represents the scatteration of point-objects in m-dimensional space (m=1, 2, 3). N is the number of the point-objects concerned. dij is distance between point i and j, while b could be 1, 2, …, and G be chosen properly case by case. When objects, on the other hand, are considered as line-objects, surface-objects and solid-objects, the formula (**) could also be used to calculate their scatteration measure. In these cases, N represents the number of points of uniform point set included within those kinds of objects. In addition, how could the formula (**) be used to