Rotate a point around another point in the Poincaré Hyperbolic Disk Suppose I have a point $P = (x_1,y_1)$ in the Poincaré disk model. How do I rotate it about another point $Q = (x_2,y_2) \neq(0,0)$ by a Euclidean angle $\alpha$? 
If $Q = (0,0)$ this is simple, just apply a normal Euclidean rotation, but I can't figure out how to do it for another point. 
Do I need to apply a translation of the disk to move $Q$ to $(0,0)$, apply the rotation, then translate back? Or is there a direct way?
 A: Found a better way :)
The idea is:


*

*Construct the hyperbolic ray $Qr$ that is ray $QP$ $\angle \alpha$ rotated around $Q$ 

*Construct  circle $e$ (hyperbolicly) centered at $Q$ through $P$

*The point you are looking for is the intersection of $e$ and $Qr$
The construction is a bit complicated:
In the construction below the elements are Euclidean elements (so lines and rays are straight Euclidean lines) also extent the elements to outside the boundary circle. 


*

*let $O$ be the centre of the Poincare disk

*Draw circle $c$  that when inside the Poincare disk represents the hyperbolic line trough P and Q

*Point $C$ be the centre of $c$

*Point  $C^1$ is point $C$ rotated $\angle \alpha$ around point $Q$

*Draw ray $q$ from $O$ through $Q$ 

*Draw line $d$ through $C$ perpendicular to $q$ (sometimes you allready have constructed this line to find point $C$)

*Draw line $s$ from $Q$ through $C^1$

*Point  $R$ is the intersection of $d$ and $s$ (so that $\angle CQR = \angle \alpha$ and $C$ and $R$ are on $d$ ) 

*Draw circle $r$  centered around $R$ through $Q$ (part of the arc inside the Poincare disk is  the ray we wanted to construct)
Then circle $e$ (hyperbolicly) centered at $Q$ through $P$


*

*Draw segment $CP$

*Draw line $p$ through $P$ perpendicular to $CP$

*Point $E$ is the intersection of $p$ and $q$ 

*Draw circle $e$  centered around $E$ through $P$ (this is the circle hyperbolicly centered around $Q$ through $P$ )

*One of the intersections of $e$ and $r$ is the point you are looking for.
DONE :)
