Simen Kvaal wrote:
>I can imagine he solution. I write this as I think, and cannot guarantee a
>working solution; only a thorough (?) analysis of the problem.
>
>Preliminary analysis: By a circle we mean not the 2d curve, but the 2d
space
>_enclosed_ by this curve, but not _including_ the curve. Thus, any point
>_inside_ a circle with radius 1 lies _closer_ than 1 units from the centre
>of the circle. The point _just outside_, however, will lie exactly one unit
>away from the centre.

Yes.

>If you have N circles (not spheres, but the 2d counterpart) in the plane,
>with radius 1, you can see this:
>
>We are seeking a point _no farther_ than 1 units away from one well chosen
>circle. This, because there are a finite number of circles, and there will
>always exist a point that is outside them all.
>
>the vector a_n which points from the centre of circle number n to the point
>P (insode) in question, has an absolute value _less_ than 1. The vector
b_n,
>pointing to the point Q _outside_all of the circles, however, has an
>absolute value _equal_ to one.

Yes, where sphere N is one specific circle, not any circle.

>Examining the perimeter of the union of the circles, we find m candidates
to
>the point Q, and these lie in the corners where circles meet. There are a
>maximum of N+1 such points, but it can be less. (I cannot prove this
easily,
>but see for yourself; it must be true).

I don't think so. If there's holes in the area covered by the spheres there
will be more points, I think.

Besides, How do I find out which intersection points are on the perimeter of
the union? I guess testing all the points are the only way?

One more question: How do I find these points where the circles intersect?

Thanks for your help!

Greetings,

Rune

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