Alignments of random points

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.

The following is a very approximate order-of-magnitude estimate of the likelihood of alignments, assuming a plane covered with uniformly distributed "significant" points.

So, based on the approximate estimates above, the expected number of k-point alignments in the overall set can be estimated to be very roughly equal to 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.

This phenomenon occurs regardless of whether the points are generated pseudo-randomly by computer, or from data sets of mundane features such as pizza restaurants or telephone booths.

On a map with a width of tens of kilometers, it is easy to find alignments of 4 to 8 points even in relatively small sets of features with w = 50 m. Choosing larger areas, denser sets of features, or larger values of w makes it easy to find alignments of 20 or more points.