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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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    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.2012-08-03

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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.

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    @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...2012-08-04