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I'm somewhat puzzled as to how to determine where a certain Möbius map takes regions. For example, consider the annuli

$$A_{1,2}:=\{z\in\mathbb{C}:1<|z|<2\}$$ $$A_{\frac{1}{2},1}:=\{z\in\mathbb{C}:1/2<|z|<1\}$$

and the Möbius map

$$ \mu(z) = \frac{(1-i)z+(1+i)}{(1+i)z+(1-i)} $$

Say, we can take three boundary points and see that they are mapped to a certain area in the upper half-plane. We also know that if the three points are not collinear, they must define a circle. But how can we determine from this the entire image of the mapping?

We can probably determine the boundaries of the regions that way. Does this mean that the rest of the points will necessarily sit between the two boundaries, since we will just have an infinite amount of circlines within those boundaries?

Would appreciate a little bit of an insight.

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Mobius transformations map circlines to circlines, so you can determine the image of each boundary circle of your annulus by looking at the image of three points from that circle. The images will divide the plane into some connected regions. Since a Mobius transformation is continuous, it maps connected regions to connected regions. So you can determine which connected region in the image space is hit by your annulus simply by taking the image of a single point from the interior of the annulus.

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    So if a Möbius map maps three points from the annulus to a circle, then does this mean that the connected region the map is mapping to is a circle with some radius? Or should this be a strip? How do I know? – 2017-02-09
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    You don't need to take the images of three points in the annulus. First take the image of three points on each boundary circle, to figure out the image of each boundary circle. This will divide the plane into three regions. Take the image of a single point in an annulus. Then the annulus maps to the connected region containing that point. – 2017-02-10