Editing Alignments of random points (section)

== More precise estimate of expected number of alignments ==
Using a similar, but more careful analysis, a more precise expression for the number of 3-point alignments of maximum width ''w'' and maximum length ''d'' expected by chance among ''n'' points placed randomly on a square of side ''L'' can be found as <ref  name=edmunds>{{Cite journal |last1=Edmunds |first1=M. G. |last2=George |first2=G. H. |date=April 1981 |title=Random alignment of quasars |url=https://www.nature.com/articles/290481a0 |journal=Nature |language=en |volume=290 |issue=5806 |pages=481–483 |doi=10.1038/290481a0 |bibcode=1981Natur.290..481E |issn=1476-4687|url-access=subscription }}</ref>
:<math> \mu = \frac {\pi } {3} \frac {w}{L} 
 \left( \frac {d}{L} \right)^3 n (n-1) 
 (n-2) </math>

If  ''d'' ≈ ''L'' and ''k'' = 3, it can be seen that this makes the same prediction as the rough estimate above, up to a constant factor.

If edge effects (alignments lost over the boundaries of the square) are included, then the expression becomes
:<math> \mu = \frac \pi  3 \frac w L 
 \left( \frac d L \right)^3 n (n-1) (n-2)
 \left( 1 - \frac 3 \pi \left( \frac d L \right) 
 + \frac 3 5 \left( \frac 4 \pi - 1 \right) 
 \left( \frac d L \right)^2 \right) </math>

A generalisation to ''k''-point alignments (ignoring edge effects) is<ref name=george>{{cite web |url=http://www.engr.mun.ca/~ggeorge/astron/thesis.html |title=Ph.D. Thesis of Glyn George: The Alignment and Clustering of Quasars |first1 = G.H |last1= George |date=2003-08-03 |accessdate=2017-02-17}}</ref>
:<math> \mu = \frac {\pi n (n-1) (n-2)  
 \cdots (n - (k-1)) } 
 {k (k-2) !} \left( \frac w L \right)^{k-2} 
 \left( \frac d L \right)^k  </math>
which has roughly similar asymptotic scaling properties as the crude approximation in the previous section, for the same reason; that combinatorial explosion for large ''n'' overwhelms the effects of other variables.