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Alignments of random points in the plane can be demonstrated by statistics to be remarkably and counter-intuitively easy to find when a large number of random points are marked on a bounded flat surface. This has been put forward as a demonstration that ley lines and other similar mysterious alignments believed by some to be phenomena of deep significance might exist solely due to chance alone, as opposed to the supernatural or anthropological explanations put forward by their proponents. The topic has also been studied in the fields of computer vision and astronomy. A number of studies have examined the mathematics of alignment of random points on the plane.〔"Alignments in Two-Dimensional Random Sets of Points" David G. Kendall and Wilfrid S. Kendall ''Advances in Applied Probability'' Vol. 12, No. 2 (Jun., 1980), pp. 380-424 Published by: Applied Probability Trust Article Stable URL: http://www.jstor.org/stable/1426603〕〔〔〔(【引用サイトリンク】title=Point Alignment Detection )〕 In all of these, the width of the line - the allowed displacement of the positions of the points from a perfect straight line - is important. It allows the fact that real-world features are not mathematical points, and that their positions need not line up exactly for them to be considered in alignment. Alfred Watkins, in his classic work on ley lines ''The Old Straight Track'', used the width of a pencil line on a map as the threshold for the tolerance of what might be regarded as an alignment. For example, using a 1 mm pencil line to draw alignments on an 1:50,000 Ordnance Survey map, a suitable width would be 50 m. == An estimate of the probability of chance alignments == Contrary to intuition, finding alignments between randomly placed points on a landscape gets progressively easier as the geographic area to be considered increases. One way of understanding this phenomenon is to see that the increase in the number of possible combinations of sets of points in that area overwhelms the decrease in the probability that any given set of points in that area line up. One definition which expresses the generally accepted meaning of "alignment" is: :''A set of points, chosen from a given set of landmark points, all of which lie within at least one straight path of a given width'' More precisely, a path of width ''w'' may be defined as the set of all points within a distance of ''w/2'' of a straight line on a plane, or a great circle on a sphere, or in general any geodesic on any other kind of manifold. Note that, in general, any given set of points that are aligned in this way will contain a large number of infinitesimally different straight paths. Therefore, only the existence of at least one straight path is necessary to determine whether a set of points is an alignment. For this reason, it is easier to count the sets of points, rather than the paths themselves. The number of alignments found is very sensitive to the allowed width ''w'', increasing approximately proportionately to ''w''''k''-2, where ''k'' is the number of points in an alignment. The following is a very approximate order-of-magnitude estimate of the likelihood of alignments, assuming a plane covered with uniformly distributed "significant" points. Consider a set of ''n'' points in a compact area with approximate diameter ''d'' and area approximately ''d''². Consider a valid line to be one where every point is within distance ''w''/2 of the line (that is, lies on a track of width ''w'', where ''w'' << ''d''). Consider all the unordered sets of ''k'' points from the ''n'' points, of which there are: : To make a rough estimate of the probability that any given subset of ''k'' points is approximately collinear in the way defined above, let us consider the line between the "leftmost" and "rightmost" two points in that set (for some arbitrary left/right axis: we can choose top and bottom for the exceptional vertical case). These two points are by definition on this line. For each of the remaining ''k''-2 points, the probability that the point is "near enough" to the line is roughly ''w''/''d'', which can be seen by considering the ratio of the area of the line tolerance zone (roughly ''wd'') and the overall area (roughly ''d''²). So, the expected number of k-point alignments, by this definition, is very roughly: : Among other things this can be used to show that, contrary to intuition, the number of k-point lines expected from random chance in a plane covered with points at a given density, for a given line width, increases much more than linearly with the size of the area considered, since the combinatorial explosion of growth in the number of possible combinations of points more than makes up for the increase in difficulty of any given combination lining up. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「alignments of random points」の詳細全文を読む スポンサード リンク
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