Determining a hyperbolic line from two points on Poincaré disk

Question

I tried to make a C++ program to make a hyperbolic line.

This is how I started:

I have two points, $P=(x_1, y_1)$ and $Q=(x_2, y_2)$, in the Poincaré disk. And the radius is 1.

$$\begin{align} \alpha &= \frac{1}{x_1^2 + y_1^2} \qquad P^{-1} = (\alpha x_1, \alpha y_1) \\[4pt] \beta &= \frac{1}{x_2^2 + y_2^2} \qquad Q^{-1} = (\beta x_2, \beta y_2) \end{align}$$

Then calculate the center of the circle as the center of the two inverted points and the radius.

But the new circle does not intersect $P$ and $Q$.

What I doing wrong?

Answer 1

Based on that site, I can make it.

http://www.ambrsoft.com/TrigoCalc/Circle3D.htm