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Given a Möbius transformation that maps one pair of concentric circles to another pair of concentric circles, why is the ratio of the radii preserved through the map?

I thought about how Möbius transformations are compositions of rotations, scaling, inversion, and translation, and that intuitively, these types of maps shouldn't change the ratio of radii between two circles.

Would it be correct to just say that if $\frac{r_1}{r_2}$ is the ratio of radii between the two circles, then

1) The radii are invariant under translation, $z \mapsto z+a$, so $\frac{r_1}{r_2}$ stays the same

2) Under scaling by a factor $z \mapsto az$, $\frac{ar_1}{ar_2} = \frac{r_1}{r_2}$

3) Under inversion, $z \mapsto \frac{1}{z}$, $\frac{1/r_1}{1/r_2} = \frac{r_2}{r_1}$

Or is there a different/better way to think about this problem?

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    Maybe you should ask for a different way to think about it rather than a _better_ way. Personally I think this is a really good way to look at it!!2011-02-25
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    Fair enough, and thanks! I guess I was just wondering if this is a standard approach, because I couldn't think of a different way to solve it.2011-02-25
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    One potential problem with (3) is that it only applies to inversion in another concentric circle. What if you invert in a different circle? Then the radii don't behave this way. On the other hand, maybe these are the only inversions you have to care about: maybe inversion in a different circle will never take the concentric circles to concentric circles. (I don't actually know but I suspect this to be the case.)2011-02-25
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    Sorry but could you clarify what you mean by inversion into a different circle (maybe an example of a specific map would help)?2011-02-25
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    An inversion operation is always with respect to some circle (or line). Inversion in a circle of radius $R$ fixes the circle, takes the center to infinity, and inverts distances to the center (so a point $r$ from the center ends up $R^2/r$ from the center). But not all distances get inverted. So if you have two concentric circles and you invert about another concentric circle, you will end up with two concentric circles, and your (3) would work. Whereas if you invert about a random circle, (3) doesn't really make sense . . . but the resulting circles probably won't be concentric anyway.2011-02-27

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