Simple formulas for geodesics of the Poincaré disk? Let $D$ be the Poincaré disk endowed with the metric $g=4\dfrac{dx^2+dy^2}{(1-x^2-y^2)^2}$. I want to find a sample equation for the geodesics between two points $p$ and $q$ in the disk. We know that the geodesics are portions of circles that intersect the boundary of the disk at right angles  and the lines through the origin. The problem is that if we transform geodesics from the Hyperbolic space the formulas becomes complicated !!
Is there any sample way to find them ?
 A: The easiest is probably to write it down in complex coordinates. A unit-speed geodesic passing through the origin $0\in{\Bbb D}$ at time zero in the direction of $n\in {\Bbb C}$ with $|n|=1$ is given by
$$ w_{n}(t) = n \tanh \frac{t}{2}.$$
It is a straight line-segment keeping the angle $\theta$ with the real axis. Now, given $p,q\in{\Bbb D}$, the Möbius transformation:
$$ w=R(z) = \frac{z-p}{1-\bar{p}z} \Leftrightarrow z=R^{-1}(w) = \frac{w+p}{1+\bar{p}w}$$
is an isometry that maps $p$ to 0. Let $q'=R(q)=\frac{q-p}{1-\bar{p}q}$ and set $n=q'/|q'|$. To get the unit speed geodesic trough $p$ and $q$ (passing $p$ at time zero) simply take the preimage of $w_n(t)$:
$$ z(t) = R^{-1}(w_n(t)).$$
You may here insert the various expressions to get a rather messy looking explicit formula (it becomes a fractional linear map in $\tanh(t/2)$ but with rather complicated coefficients).
Depending on what you want to do with the geodesic you might not want to use this explicit formula.
Möbius transformations preserves the collection of disks union lines as well as angles (being conformal) so as you mention the trajectory of  $z(t)$ must be part of a circle perpendicular to the unit circle.
