Complex mapping z ↦ ω = (z − a)/(z − b)

[Hill]

Homework Statement

Consider the complex mapping $z \mapsto \omega = \frac {z - a} {z - b}$. Show geometrically that if we apply this mapping to the perpendicular bisector of the line-segment joining a and b, then the image is the unit circle. In greater detail, describe the motion of ω round this circle as z travels along the line at constant speed.

Relevant Equations

geometry

All points on that line are equidistant from the points a and b. Thus, the length of $\frac {z - a} {z - b}$ is 1, i.e., the points on the unit circle.
If the angle of $z - a$ is $\alpha$, and the angle of $z - b$ is $\beta$, then the angle of $\frac {z - a} {z - b}$ is $\alpha - \beta$. If the point z is far toward either end of the line, $\alpha - \beta$ is close to zero. If z is in the midpoint between a and b, $\alpha - \beta = \pi$. Thus, as z travels along the line from one "end" to another, w starts near 1, moves along one half of the unit circle speeding up towards -1, and then slows down again as it continues on the other half of the unit circle toward 1.
Are there other geometric details that I've missed?

 

[FactChecker]

This is good, assuming that you have not already studied these transformations before and are supposed to know established theorems for them. They are called "bilinear transformations" or "Mobius transformations". I can not think of anything that you have missed.

PS There is a lot more that can be said about these transformations, but I don't think they will help to prove what you proved.

 

[Hill]

This is good, assuming that you have not already studied these transformations before and are supposed to know established theorems for them. They are called "bilinear transformations" or "Mobius transformations". I can not think of anything that you have missed.

PS There is a lot more that can be said about these transformations, but I don't think they will help to prove what you proved.

Thank you. I see that the next chapter in the textbook is "Möbius Transformations and Inversion."

 

[WWGD]

There's an interesting result to the effect that a Mobius Transformation is uniquely determined by the image of 3 of its points
https://math.stackexchange.com/ques...determined-based-on-known-mappings-of-three-p

 

[Hill]

There's an interesting result to the effect that a Mobius Transformation is uniquely determined by the image of 3 of its points
https://math.stackexchange.com/questions/1011697/why-is-a-möbius-transform-uniquely-determined-based-on-known-mappings-of-three-p

The Mobius Transformation is the next chapter in the book. I assume the author will refer to the exercises of this chapter then.

 

[FactChecker]

I personally liked and was impressed by your proof. I think it is the right kind of reasoning to understand analytic functions of a complex variable. Of course, if you have theorems that can be used, that is the first thing to look for. But without applicable theorems, your kind of analysis is good.