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I need to find the equation for the figure defined by several intersecting circles. Finding the total area is not important, but knowing the bounds of the figure is, because I want to find the point within the figure that optimizes something.

The first step would be to determine if there even is an intersection of all my circles. Is there a way to test this for any number of circles on the Cartesian plane, where the $i$th circle has radius $r_i$ and center $(a_i,b_i)$?

If so, is there a way to further know the boundaries of that intersection, so that I can choose a point within it according to some optimization criterion?

The optimization criterion is actually the point in the intersection that minimizes the distance between the point and any point in another circle, one that doesn't intersect with any of the first group of circles.

Any suggestions on how to approach this? I believe if I combine $(x_i-a_i)^2 + (y_i-b_i)^2 = r_i^2$ with $(x_{i+1}-a_{i+1})^2 + (y_{i+1}-b_{i+1})^2 = r_{i+1}^2$, I'll be able to find an equation that defines the area, but then I don't know how to use that to define a range I can choose among for my optimization.

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There are efficient methods to find the convex hull of a set of points that could likely be adapted for finding which circles are on the boundary of a mass of circles. Some methods might require that the circles are at least roughly similar in size. Similar algorithms can find the Voronoi/Dirichlet tessellation, which could be useful for deciding where to check centre-to-centre distances for overlap.

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