# Constructing a point on a geodesic line in the hyperbolic disc with a given distance from another point

Given a geodesic line $$L$$ in the Poincaré disc $$\{ (x,y) \in \mathbb{R}^2 : |x|^2 + |y|^2 < 1 \}$$ with the hyperbolic metric and a point $$P \in L$$, how can one construct a point $$Q \in L$$ that has a given hyperbolic distance $$D > 0$$ from $$P$$?

Note that if we denote the ideal endpoints of $$L$$ by $$A$$ and $$B$$, then the hyperbolic distance between points $$P, Q \in L$$ is given by $$d(P,Q) = \ln \Bigl(\frac{|AQ| \cdot |PB|}{|AP| \cdot |QB|} \Bigr).$$ But I don't see how this leads to a construction of $$Q$$.

You don't really use that distance formula to find the two points (if $$D > 0$$) on $$L$$ that are the distance $$D$$ from $$P$$. You use the fact that a hyperbolic circle appears in the Poincare disk model as a Euclidean circle (but with displaced center).
On the disk's diameter passing through $$P$$, find the two points having the distance $$D$$ from $$P$$ along that diameter. (If $$D/2 > \arctan(r)$$, where $$r$$ is the Euclidean distance from the center of the unit disk model to the point $$P$$, one of these two points is on the other side of the center.) This is a diameter of the circle of hyperbolic radius $$D$$ centered at $$P$$. Now draw the Euclidean circle with that diameter. That circle intersects $$L$$ in two points, both of which are distance $$D$$ from $$P$$ along $$L$$.
For computation, it can be useful to know: For two points, $$x = (r,\theta)$$ and $$x' = (r',\theta)$$ with $$0 \leq r' < r < 1$$, in Euclidean polar coordinates on the same ray from the origin of the unit circle Poincare model, their hyperbolic distance is $$\ln \left( \frac{1+r}{1-r} \cdot \frac{1-r'}{1+r'} \right) = 2(\arctan r - \arctan r') \text{.}$$ (In the case where the second point is on the other side of the center, use this formula to get the hyperbolic distance from $$P = (r,\theta)$$ to the center (at $$(r' = 0,\theta)$$), then find the point, $$(r,-\theta)$$, on the other side of the origin, $$(r'=0,-\theta)$$, having the rest of the distance.)