Editing Alignments of random points (section)

{{short description|Phenomenon in statistics}}
[[Image:Ley lines.svg|thumb|300px|right|80 4-point near-alignments of 137 random points]]

The study of '''alignments of random points in a plane''' seeks to discover subsets of points that occupy an approximately straight line within a larger set of points that are [[randomness|randomly]] placed in a planar region.
Studies have shown that such near-alignments occur by chance with greater frequency than one might [[counter-intuitively|intuitively]] expect.

This has been put forward as a demonstration that [[ley line]]s 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.<ref>{{Cite journal |last1=Kendall |first1=David G. |last2=Kendall |first2=Wilfrid S. |date=1980 |title=Alignments in Two-Dimensional Random Sets of Points |url=https://www.jstor.org/stable/1426603 |journal=Advances in Applied Probability |volume=12 |issue=2 |pages=380–424 |doi=10.2307/1426603 |jstor=1426603 |issn=0001-8678|url-access=subscription }}</ref><ref name=edmunds/><ref name=george/><ref name=hal-inria>{{Citation |last1=Lezama |first1=José |title=A Contrario 2D Point Alignment Detection |date=2014-03-06 |url=https://inria.hal.science/hal-00956596 |access-date=2023-09-29 |language=en |last2=Gioi |first2=Rafael Grompone von |last3=Morel |first3=Jean-Michel |last4=Randall |first4=Grégory}}</ref> 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&nbsp;mm pencil line to draw alignments on a [[Scale (map)|1:50,000 scale]] [[Ordnance Survey]] map, the corresponding width on the ground would be 50 m.<ref>{{cite book |isbn = 9780349137070 | title = The Old Straight Track: Its Mounds, Beacons, Moats, Sites and Mark Stones |last1 = Watkins |first1 = Alfred |date = 1988 |publisher = Abacus  }}</ref>