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I have a set of N circles on a 2-D plane. These N circles all contain the same point (coordinates unknown) so there is a common union between all the circles. How could I go about finding the center of the union of the N circles?

Any help would be appreciated


alt text
I am looking for the red dot in each.

-I know the coordinates of the centers of the circles and their radii. -Circles can be different sizes

So no, I do not think center of mass will work.

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closed as off topic by Michael Renardy, Kevin Walker, Emil Jeřábek, Andres Caicedo, Will Jagy Apr 24 '13 at 19:33

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For one thing, you need to define "center" for something that is not a circle. I presume you mean the center of gravity. Now consider what happens if you concentrate the mass of each circle at its midpoint ... This site is for research level questions. Voting to close. –  Michael Renardy Apr 24 '13 at 16:53
I could be totally wrong, but the wording makes me suspect that the OP might actually mean an intersection of disks rather than a union of circles, in which case it is not so trivial. –  Emil Jeřábek Apr 24 '13 at 17:10
From the given data you cannot find the center of gravity of the set of circles. There are infinitely many possible configurations of N circles with a common point. –  Rhett Butler Apr 24 '13 at 17:12
I take back my earlier comment. I think the way the problem is intended, the mass density where the circles overlap is supposed to be 1 rather than adding up the masses of the circles. With that interpretation, the problem does not seem trivial. –  Michael Renardy Apr 24 '13 at 17:32
Although the question is not clear as presented, one interpretation is, I think, nontrivial and not answered by the comments: Compute the center of gravity of the region determined by the intersection of $n$ given disks (given by centers and radii). And compute efficiently. –  Joseph O'Rourke Apr 25 '13 at 0:09

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