Given a line in hyperbolic geometry construct a point symmetric to a given point.

Question

Please help me to solve the following problem:

Use upper half-plane model of hyperbolic geometry. Prove that inversion across a line $L$ is an isometry and construct a point $P'$ that is symmetric to a given point $P$ across $L$.

The distance between points $A$ and $B$ is the following : $$d(A, B) = |\ln[A,B,X,Y]|,$$ where $[x_1,x_2,x_3,x_4]$ is the cross-ratio. Lines in hyperbolic geometry are the following: vertical lines and half-circles centered on the absolute. $L'$ is the line through $A$ and $B$. $X$ and $Y$ are points of $L'$ on the absolute. If $L'$ is vertical, then $Y=\infty$. Order of points on $L'$ is $Y, A, B, X$.

My attempt:

1. I was able to prove that if line is not vertical, then any inversion has the form $z\rightarrow\frac{a\overline{z}+b}{c\overline{z}+d}$. I know a bit more about $a,b,c,d$ since it is not arbitrary inversion, but it does not matter. I failed to understand what the inversion is if line is vertical.

2. I was able to prove that $z\rightarrow\frac{a\overline{z}+b}{c\overline{z}+d}$ "almost" keeps the cross-ratio - there result is complex conjugate of cross-ratio for normal Mobius transformation.

3. Now it should be immediate that the mapping is isometry, because it looks like $\ln(z) = \ln(\overline{z})$. But unfortunately I do not understand the concept of complex logarithm good enough. Is true? How to prove it?

4. I believe that symmetric point for non-vertical line is the image under inversion. I was not able to prove it. Is there any way to prove it without messy calculations using $d(A,B)$? If not, then how to do it with $d(A,B)$.

5. I do not know what is symmetric point for vertical line. Is it "normal" symmetrical point? How to prove it?

Thanks a lot for your hints and your answers!

Answer 1

1). A reflection in a vertical line is in the form of $ - \overline{z}+b $ with $b \in R$ (thus $a=-1 , c=0 , d=1 $)

3). I think you made a mistake with $Y$ It is not $\infty $ but $Y= a + \infty i $ (Y point on the line x=a) the distance on a vertical line is $| \ln (a) -\ln (b) |$ should work now

Also the cross ratio is about the distances between some points not about the (complex) difference between when taken them as complex numbers (I made a new question about this see xxxxx )

4) needs further investigation I think you need messy distances

5) Yes

Answer 2

Comment/Observation

I have proposed the external black circle which is a pure inversion to the red circle with respect to mirroring/inverting circle centered at $O$ as a valid extension to the Poincare disc model, that had not been specifically mentioned so by him..

Is it relevant to what you are asking?