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jwellens
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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 $s$?

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 of their union?

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 $s$?

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jwellens
jwellens
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  • 8

Area of intersectionunion of random circles in a plane

It began to rain as I was standing outside today, and while watching the raindrops wet the cement I thought of the following question:

Assuming $n$ raindrops (i.e. circles of radius $r$) fall randomly, independently and uniformly over a square of area $A$, what is the expected area of the wet region?

As $r/A$ approaches zero, the expected area should approach $\pi r^2 n$, but when the raindrops aren't tiny, the probability of drops intersecting isn't negligible. Also, there are edge effects because the square is finite, but I'm not so interested in those. Actually, the problem could be generalized and restated as follows:

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 of their union?

Area of intersection of random circles in a plane

It began to rain as I was standing outside today, and while watching the raindrops wet the cement I thought of the following question:

Assuming $n$ raindrops (i.e. circles of radius $r$) fall randomly, independently and uniformly over a square of area $A$, what is the expected area of the wet region?

As $r/A$ approaches zero, the expected area should approach $\pi r^2 n$, but when the raindrops aren't tiny, the probability of drops intersecting isn't negligible. Also, there are edge effects because the square is finite, but I'm not so interested in those. Actually, the problem could be generalized and restated as follows:

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 of their union?

Area of union of random circles in a plane

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 of their union?

Source Link
jwellens
jwellens
  • 413
  • 2
  • 8

Area of intersection of random circles in a plane

It began to rain as I was standing outside today, and while watching the raindrops wet the cement I thought of the following question:

Assuming $n$ raindrops (i.e. circles of radius $r$) fall randomly, independently and uniformly over a square of area $A$, what is the expected area of the wet region?

As $r/A$ approaches zero, the expected area should approach $\pi r^2 n$, but when the raindrops aren't tiny, the probability of drops intersecting isn't negligible. Also, there are edge effects because the square is finite, but I'm not so interested in those. Actually, the problem could be generalized and restated as follows:

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 of their union?