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.)

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