Sorry for the awful title, but this is a difficult problem to describe, so I made a picture. I want to find R given theta and all of the outer radii. Each of the outer circles must be centered on the circumference of the inner circle, and angle separating each of them (from their intersection with the larger circle) must be theta. The image illustrates the problem with 5 outer circles, but I need a solution that is generalized for n outer circles. How can I find R as a function of theta and all of the smaller radii?

I've done some math and come up with the following equation (and I can explain its basis if that would help):

$$ 2 \pi = n\theta + \sum_{i=0}^{n} 4 \arcsin{\frac{r_i}{2R}} $$

But as far as I know it is impossible to solve for R in any generalized way. Could someone give me some pointers? Thanks.

  • $\begingroup$ I am pretty sure there's no closed form solution to that equation. Cut and try (or perhaps mathematica) could give you a numerical solution. $\endgroup$ – Ethan Bolker Apr 7 '16 at 19:28
  • $\begingroup$ Yeah, I feared that. If this equation has no closed-form solution, does that mean that it is impossible for the problem to have a closed-form solution (if someone came up with a different approach)? $\endgroup$ – sgfw Apr 7 '16 at 19:30
  • $\begingroup$ I think there are a few typos in your equation. Presumably it is $nR\theta$ not $n\theta$ and $\sum 4R\dots$ not $\sum 4r_i\dots$ $\endgroup$ – almagest Apr 7 '16 at 19:35
  • $\begingroup$ I suspect no approach will lead to a closed form solution. Might an approximation do? If you can say why you need to know there may be another way to attack your problem - see meta.stackexchange.com/questions/66377/what-is-the-xy-problem $\endgroup$ – Ethan Bolker Apr 7 '16 at 19:36
  • $\begingroup$ Thanks for pointing out the typos. I fixed them and then canceled out the Rs on both sides to simplify it a bit. $\endgroup$ – sgfw Apr 7 '16 at 19:39

Taking sine of both sides of your equation leads to a complicated polynomial equation in $R$ and $\sqrt{4R^2 - r_i^2}$. In principle this means $R$ is an algebraic function of the parameters, but a "closed-form" solution is not to be expected.

I would start with the approximation $\arcsin (r_i/(2R)) \approx r_i/(2R)$, which is good when $R$ is large compared to $r_i$, leading to $$ R \approx R_0 = \dfrac{2}{2\pi - n \theta} \sum_i r_i$$ and then use a few iterations of Newton's method to refine it:

$$R_{k+1} = R_k + \dfrac{R_k \left(n \theta - 2 \pi + 4 \sum_i \arcsin(r_i/(2R_k))\right)}{4 \sum_i \left(r_i/\sqrt{4 R_k^2 - r_i^2}\right)}$$


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