# How to find the intersection of union of two circle groups

I have two groups of circles. S1 is the union of the first group and S2 is the union of the second group of circles. I know center and radius of all circles. I have to find the equation for the intersection of S1 and S2. Is there anyway to do it?

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For the sake of searchability, "circle group" has a different and precise mathematical definition (en.wikipedia.org/wiki/Circle_group) so it might be better to say something like "collection of circles." – Qiaochu Yuan Aug 3 '12 at 21:08
What do you mean by the "equation" for the intersection? The area? – mjqxxxx Aug 3 '12 at 21:12
I mean any mathematical representation of the area that i am looking for. – kotoll Aug 3 '12 at 21:20
The description you gave (the intersection of the respective unions of two known collections of circles) is already a mathematical representation of a subset of the plane. Whether there is another, more useful, representation depends on what you want to do with it. If you want to test whether a given point belongs to the intersection, that description is probably fine. If you want the perimeter or the area or the number of disjoint closed components, then not so much. – mjqxxxx Aug 3 '12 at 22:01

Intersection of unions is the union of intersections (intersection is distributive over union): $$\left( \bigcup_{c\in \mathcal S_1}c\right)\cap \left(\bigcup_{c\in \mathcal S_2}c\right)=\bigcup_{c_1\in \mathcal S_1,c_2\in\mathcal S_2}(c_1\cap c_2)$$ So you can just intersect individual circles and take union of the intersections.
@kamuran: you don't select them; you intersect every single pair. That's what the union means: you take union over all pairs. You can write it as $\bigcup_{(c_1,c_2)\in \mathcal S_1\times \mathcal S_2}$, if you prefer... – tomasz Aug 4 '12 at 1:06