Uncovering the Error in Finding the Mobius Transformation for a Circle Mapping

[mimsy57]

I'm looking for the error in my understanding here, not help with the problem itself. I'm making some kind of mistake, so I've listed out everything I think I know, and I'm hoping someone can either tell me what I'm misunderstanding, or tell me there is an error in the problem statement:

Problem Statement: find the Mobius transformation taking the circle |z|=1 to |z+2|=1 such that T(-1)=-3 and T(i)=-1.

What I think:

*Both the circle and its image have radius 1, with the first centered at the origin and the second at -2.

*The additional points we are given mappings for are -1 and i, which are the endpoints to an arc of a quarter circle. These are mapping to -3 and -1 which are the bounds for the half circle. This implies the mapping is going around twice for once around the circle being mapped, which would imply it is not a bijection, and Mobius transformations are bijections.

 

[Dick]

I'm looking for the error in my understanding here, not help with the problem itself. I'm making some kind of mistake, so I've listed out everything I think I know, and I'm hoping someone can either tell me what I'm misunderstanding, or tell me there is an error in the problem statement:

Problem Statement: find the Mobius transformation taking the circle |z|=1 to |z+2|=1 such that T(-1)=-3 and T(i)=-1.

What I think:

*Both the circle and its image have radius 1, with the first centered at the origin and the second at -2.

*The additional points we are given mappings for are -1 and i, which are the endpoints to an arc of a quarter circle. These are mapping to -3 and -1 which are the bounds for the half circle. This implies the mapping is going around twice for once around the circle being mapped, which would imply it is not a bijection, and Mobius transformations are bijections.

Sure you can do it. You can map any three points in the complex plane to any other three points with a Mobius transformation, right? Just pick a third point on each of the two circles and that will fix a transformation. I haven't tried to visualize it in terms of dilations, rotations, etc.

 

[mimsy57]

Yes about the three points. My problem isn't so much with that, I'm okay calculating it, I'm just bothered by how it can work.

If I translate a circle, rotate a circle, or dilate a circle, or invert a circle, I don't see how points on a quarter arc could move to a half arc. Does my visualization problem make sense? I have to be understanding something incorrectly. Maybe inversion since the other three are so simple to visualize.

I tried finding examples, but in every example, the points stay in the same relative (for lack of a better word) location

 

[Dick]

Yes about the three points. My problem isn't so much with that, I'm okay calculating it, I'm just bothered by how it can work.

If I translate a circle, rotate a circle, or dilate a circle, or invert a circle, I don't see how points on a quarter arc could move to a half arc. Does my visualization problem make sense? I have to be understanding something incorrectly. Maybe inversion since the other three are so simple to visualize.

I tried finding examples, but in every example, the points stay in the same relative (for lack of a better word) location

Yes, it's inversions. Replace |z+2|=1 with, say |z+100|=99. It should be visually clear that most of the points on the circle map to points near 0. z=(-1) and points nearby don't. It stretches angles on the circle. It's a good thing to visualize, keep it up!

 

[mimsy57]

Thanks!