Circle Covering by Arcs

arc length α

0.1

number of arcs n

5

random seed

If

n

points are chosen at random on a circle with unit circumference, and an arc of length α is extended counterclockwise from each point, then the probability that the entire circle is covered is

P(α,n)=

⌊1/α⌋

∑

k=0

k

(-1)



n

k



(n-1)

(1-kα)

, and the probability that the arcs leave

l

uncovered gaps is



n

k



⌊1/α⌋

∑

j=l

j-l

(-1)



n

k



n-1

(1-jα)

. These results were first proved by L. W. Stevens in 1939. In the image, you can adjust α and

n

and compare observed circle coverings to the theory. Note that, especially when the arc length is small, there is a reasonable chance that some of the uncovered gaps will be too small to see.