I have two disks $(x-a_1)^2+(y-b_1)^2\leq r_1^2$ and $(x-a_2)^2+(y-b_2)^2\leq r_2^2$, where $a_1$, $b_1$, $r_1$, $a_2$, $b_2$, $r_2$ are all known. What kind of constraint can I put on $a_i$, $b_i$ and $r_i$ that the intersection of two disks is inside unit circle? The question is for intersection of two disks, but the generalization for $n$ disks would be even better.
How to constrain disks that intersection of them is inside unit circle
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algebra-precalculus
analytic-geometry
circles
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1unit circle centered at any particular point? – 2012-09-02
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0I guess it doesn't matter, but, maybe you want $r_1^2$ and $r_2^2$ for the sake of geometric clarity. – 2012-09-02
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0@Sasha, i have to define $a_i$, $b_i$ and $r_i$, so i think they can be centered any point – 2012-09-02
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0@JamesS.Cook equations are edited. – 2012-09-02
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0I am still looking for suggestions? – 2012-09-21
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1You might be able to simplify the problem with a well-chosen Möbius transformation. Maybe try transforming the unit circle into a line, or transform the two disks so that they have the same radius. – 2013-07-29
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0The two deleted answers fail on the same counterexample, so I'll leave it here for reference: two disks centered at $(3,0)$ with radius $3$ and at $(2,0)$ with radius $2.01$ respectively. This shows that it is not sufficient to ensure that the intersection of the *circles* bounding the disks lies within a unit circle. – 2013-08-04