I want to find an expression for r in terms of the other parameters mentioned. This algorithm will then be used inside an microcontroller , to control the existing machinery
Best Regards TaimoorAli
2026-03-29 14:28:48.1774794528
Analytical Expression for a radius in the circle geometry figure shown
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If I understand you correctly, the triangle formed by the centres of the three circles has sides $r+r_2$, $r+r_3$ and $a$, with the angle opposite the side $a$ being $\theta = 2 \pi - \text{arc}/r$. The Law of Cosines gives you an equation
$$ a^2 = (r+r_2)^2 + (r + r_3)^2 - 2 (r+r_2)(r+r_3) \cos(\theta)$$ However, since $\theta$ is a rational function of $r$, there is no way you're going to get a closed-form solution for $r$ as a function of $r_1, r_2, a, \text{arc}$. You'll need numerical methods.