Distance in Poincaré disk from origin to a point given

Question

Let $C$ circle $x^2+y^2=1$ find the distance (Poincaré disk) from $O=(0,0)$ to $(x,y)$

The distance in Poincaré is $d=ln(AB,PQ)$ where AB are a segment of the curve and P and Q are points in the limits of Poincaré disk. Then $A=(0,0)$ and $B=(x,y)$ but I dont know the values of P and Q. I try to use the circunference formula, but I have only two points (A and B) and I need three. Please give me clues, to solve this problem. Thank you.

Answer 1

See, metric entry in the Poincaré Disk Model :

$$|u| = \sqrt{x^2 + y^2}$$ $$\delta(u, v) = 2 * \frac{|u-v]^2}{(1 - |u|^2)(1 - |v|^2)}$$

In your case, $v = O$ and $|v| = 0$ so : $$\delta(u, O) = 2 * \frac{|u|^2}{(1 - |u|^2)}$$ $$argcosh(1 + \delta(u, O))$$

Answer 2

In the Poincare disk, "lines" are circle or line segments that intersect the unit circle in two points, and which are perpendicular to the unit circle at the points of intersection. In particular, "Lines" through the origin are diameters of the unit circle, while all "lines" that miss the origin are circle segments.

However, in your case, you are talking about a line through the origin, So $P$ and $Q$ are simply the endpoints of the diameter that passes through $B = (x, y)$.