Details

The direction of the force of gravity is determined by the angle , in , such that the swarm can assume eight different polygonal shapes. The shapes alternate between triangles and trapezoids when , and alternate between squares with one corner removed and trapezoids when .

Computing the means and , variances and , covariance , and correlation requires integration over the area containing the swarm. One way is to use an indicator function that returns 1 if the point is inside the region containing the swarm and 0 otherwise. The formulas are as follows, integrating over the unit square with and from 0 to 1.

, ,

, ,

,

.

Instead of using an indicator function, the region of integration can be changed to only include the polygon containing the swarm. As an example calculation, if the force angle is , the mean when the swarm is in the lower-left corner is

for and for .

A few interesting results: the correlation is maximized when the swarm has a triangular shape, and equals . The covariance of the triangle is always . Variance is maximized in one direction and minimized in the other when the swarm is in a rectangular position. Mean positions are maximized when is small.