3

enter image description here

I tried to do draw the problem in 3D but I stil can't imagine how such sphere can exist... it's really confusing

Plato
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2 Answers2

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Let us first find out which spheres "fit snugly" against the arc between $A$ and $B$. Such a sphere must obviously fit against the whole circle of which the arc is a part. This means that the center of the sphere must be on the symmetry axis of the circle (the line normal to the circle's plane that goes through its center). If $D$ is placed at the origin (and other points are treated as vectors), this line goes through the point $A+B$ in the direction $C$.

Similar considerations of course can be made for the other two arcs. We obtain three lines, and the center of the sphere should be on all of them (because the sphere should fit against all arcs). Fortunately these lines do intersect, and this happens at the point $A+B+C$. This is the opposite point of $D$ in the cube which has the edges $DA$, $DB$ and $DC$.

The radius you need is now the distance from this point to any of the points $A,B,C$. It should now be simple enough to finally figure out the radius. (Do ask if clarification is needed.)

Joonas Ilmavirta
  • 25,809
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Form a cube with A, B, C and D at its corners. The centre of the sphere is at the corner opposite D. From that you should be able to work out its radius.

enter image description here

Jyrki Lahtonen
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MartinG
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  • Also, what is the motivation to choose D? – Plato Sep 08 '14 at 18:08
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    I think this turns out to be a valid approach, although one has to work a little to be sure. – rschwieb Sep 08 '14 at 18:35
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    Looking at these dice may help. – MartinG Sep 08 '14 at 20:40
  • The point I described is equidistant from every point on each of the three circles, since it's where their axes of symmetry meet. It's therefore the centre of a sphere on which these circles lie. I think that's clear - and maybe worth an un-downvote? :) – MartinG Sep 08 '14 at 22:12
  • Martin, I took the liberty of adding a pic. Less stylized but perhaps clearer than the dice in your link. Do remove it if you don't want it :-) Also, people are unable to "un-downvote" unless the post is edited. The vote is locked after a few minutes. May be you didn't know that? – Jyrki Lahtonen Sep 09 '14 at 14:44