I'm realizing how little (in some respects) I know about circles. Here's something that emerged out of something I was fiddling with. My question is whether this is
- "well known" in the way that $229\times983=225107$ is "well known" (don't publish it unless you're publishing a table); or
- well known in the sense that every book includes it (for suitable values of "every"); or
- well known in the sense that everybody knows it (for at least moderately reasonable values of "everybody").
I'm looking at the circle $|z|=1$ in $\mathbb{C}$. Let $$f(z)=\dfrac{-3z+1}{z-3}.\tag{This is $f$.}$$ This of course fixes $\pm 1$ and leaves the circle invariant, and maps $\pm i$ to $\dfrac{-3\pm4i}{5}$. If we draw a circle through those two images of $\pm i$ meeting the unit circle at a right angle, it is centered at $-5/3$ and has radius $4/3$. That circle meets the real axis at $-1/3$. So look at the line $\operatorname{Re}=-1/3$. Look at the point on that line where $\operatorname{Im} = y$. Draw the line through that point and the aformentioned center $-5/3$. That line crosses the circle twice. It would seem that those two points are $f(z)$ and $f(-\bar z)$, where $z$ and $-\bar z$ are the two points on the unit circle with imaginary part $y$.
This gives us a simple geometric picture of how $f$ behaves. That allows us to use routine Euclidean geometry to show that $\theta\mapsto f(e^{i\theta})$ satisfies the differential equation
$$
\left|\dfrac{dg}{d\theta}\right| = \text{constant}\cdot\operatorname{Re}\left(g-\left(-\dfrac 5 3\right)\right)
$$
subject to the constraint that the values of $g$ are on the unit circle. (The equation says the rate at which $g$ moves along the circle is proportional to a certain affine function of the real part.)
Best Answer
In my opinion:
I don't believe your construction is "well-known" in any of your three versions of "well-known". I am pretty sure that it is the sort of thing that could appear as an exercise in any of the well-known classical texts on complex analysis, but I am pretty sure I have never done anything quite like this, despite having done many exercises from standard texts.
A very quick search through some texts (Ahlfors, Burckel, Lang, Conway) shows nothing quite like it.
I suspect (without doing any analysis yet) that your results depend on the fact that your transformation is of the form $z\mapsto\frac{z-a}{1-\bar{a}z}$, and would be keen to hear if you have investigated further.