Area of Union of n circles

Question

I am trying to calculate the area of union of n circles in a plane when it is known that all circles are of equal radii and their centers are also known(of all n circles). I was trying to follow the set theory approach(inclusion-exclusion principle), where we know the formula for union of n sets. I was just using an operator Ar() which gives the area,i.e. Ar(A) gives me the area of A. I first tried to find out which circle is intersecting with which other circle(s) with the help of D<2R(D=dist between the centers of the two circles), then I was trying to calculate the area of intersection between them pairwise and hence find the area of union. But I am getting stuck for n>4. Can anyone provide a soln to this(soln by the set theory approach is necessary). Thanks in advance

Answer 1

For the inclusion-exclusion approach, you need to be able to calculate for each set $S$ of circles the area $A_S$ of their intersection. Consider a set of circles, all of radius $1$, whose intersection is nonempty. The intersection will be a convex region bounded by $k$ arcs (where $k$ might be less than the number of circles); ignoring trivial cases, I'll suppose $k\ge 2$. Let $P_i = (x_i, y_i), i=0 \ldots k$, be the endpoints of the arcs, taken counterclockwise, with (for convenience) $P_0 = P_k$. Note that the area of the "cap" cut off from a circle of radius $1$ by a chord of length $L$ is $f(L) = \arcsin(L/2) - L \sqrt{4 - L^2}/4$, while the area of the polygon with vertices $P_i$ is $\sum_{i=1}^k (x_{i-1} - x_i)(y_{i-1}+y_i)/2$. So the total area of the intersection is $$A_S = \sum_{i=1}^k \left( f\left(\sqrt{(x_i - x_{i-1})^2 + (y_i - y_{i-1})^2}\right) + \frac{(x_{i-1} - x_i)(y_{i-1}+y_i)}2 \right)$$