Details

Though rectangular boundaries are common in artificial workspaces, biological workspaces are usually rounded. The workspace in this example is a circle centered at with radius 1 and thus area . For notational simplicity, the swarm is parameterized by the angle of the force of gravity and the fill level . The range of possible angles for the angle of the force of gravity is . In this range of angles, the swarm always fills the area under a chord, but this shape is rotated around the unit circle.

The area under a chord of a circle is the area of a sector less the area of the triangle originating at the circle center:

,

thus

,

where is the arc length, is the chord length, is the radius, and is the height. Solving for and gives

,

.

Therefore, the area under a chord is

.

For a circular workspace with given , the mean of and is:

,

.

The variance of and is:

,

,

and finally,

.

The locus of mean positions is aligned with , and the mean position is at radius from the center. The variance is maximized at and , while the covariance is maximized at and . For small values, the correlation approaches .