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Let a circle with center $b$ and radius $r$ be contained in a circle with center $a$ and radius $R$.

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Given a point $c$ on the small circle, find its distance to the greater circle.
That is find the length of $cd$.
Assume
- the length $ab$, $r$ and $R$ are known,
- the $\mathbb R^2$ coordinates of $a$, $b$ and $c$ are known.

Using the dot product, I can find the angle $abc$.
But, then the cosine law does not uniquely determined the length $bd$. $$ r + cd = \left(ab\right) \cos abc \pm \sqrt{R^2 - \left(ab\right)^2 \sin^2 abc}. $$

Is there a way to uniquely determine $cd$?

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1 Answer 1

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Hint: The minimum (and maximum) distance between a point on the smaller circle and the closest (and farthest) point on the larger circle occurs when the tangent planes to each point on both circles are parallel. You can use Lagrange multipliers to show this but there may be simpler ways.

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  • $\begingroup$ If I understand correctly, I draw the tangent $t$ at the point $c$. Then I draw a tangent $T$ of the great circle parallel to $t$. Let say $T$ is on the side of the point $d$. Then the distance between the tangencial contact points is the minimum distance from the point $c$ to the great circle. I don't see how this resolve the uniqueness of the length $cd$. $\endgroup$
    – user14108
    Jun 7, 2014 at 0:13
  • $\begingroup$ Also the tangent $T$ does not pass through the point $d$. And the distance $cd$ seems shorter than the distance $tT$ (at the tangencial contact point). $\endgroup$
    – user14108
    Jun 7, 2014 at 0:16

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