Let a circle of radius $r$ be situated inside another circle of radius $R$. Let $d$ the distance between the centers of these circles, so $0\le d< R-r$. It is known that there exists a Möbius transformation transformation that takes these two circles to a pair of concentric circles. While the transformation is not unique, the ratio of the radiuses of these two concentric circles is uniquely determined, so a function of $R$, $r$ and $d$. Could anyone provide a formula for this ratio? References would be very helpful too. Thanks!
2026-03-27 23:20:10.1774653610
Conformal modulus of a region bounded by $2$ circles.
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