Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

If $n$ circles of radius $r$ are drawn at random in the plane such that their centers lie in some region $S$ (we could take $S$ to be a square), what is the expected area $E$ of their union?

Edit: In light of Noam's comment, it seems like the answer is

$\displaystyle \int \int 1- (1-f(x,y))^n dx dy$

where $f(x,y) = a(B_r(x,y) \cap S)/a(s)$; the probability a random circle contains $(x,y)$.

If the region $S$ is large compared to $r$ and it has a reasonably "out of the way" boundary, then $f(x,y)$ is approximately constant over all of $S$, and is about $\pi r^2/ a(S)$, and so the expected value $E$ for a fixed $n$ is $n\pi r^2 + O(r^4)$.

However, the more interesting cases occur when $r$ is not small or when $\partial S$ gets "in the way" significantly.

Suppose $S$ is a square of side length $s$ and the circles have radius $r=s/2$. How does $E$ grow with $n$ asymptotically? What if $S$ is a circle of radius $s$? A regular $k$-gon inscribed in a circle of radius $s$?

share|improve this question
5  
Compute the probability $p$ that a given point of $S$ is not hit by any of the circles (which is constant away from the edge), and multiply $1-p$ by the area of $S$. –  Noam D. Elkies Sep 3 '12 at 15:58
1  
@Noam's comment is obviously the answer, with the one caveat that a non-asymptotic result will be a little tricky. –  Igor Rivin Sep 3 '12 at 16:13
    
You will probably get a more "natural" answer if you choose a "torus", i.e., identify opposite edges of a square, to eliminate edge effects. –  Dick Palais Sep 3 '12 at 23:24
add comment

1 Answer

The question is already interesting for general $r$ when $S=[0,1]$. Then the area of $E$ is $\pi r^2 + 2 r$. Because the area defect to $E$ of two circles being $d$ apart is $\frac{d^3}{12~ r}$, and $d\approx n^{-1}$, for large $n$ only the distance between minimum $a$ and maximum $b$ of the circle centers in $S$ determines the asymptotics. The independent expectation values for $a$ and $b$ are $1/(n+1)$ and $1-1/(n+1)$ giving an asymptotic area defect to $E$ of $4r/(n+1)$.

share|improve this answer
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.