  We consider Jordan curves $\gamma=\cup_{j=1}^n \gamma_j$ in the Riemann
sphere for which each $\gamma_j$ is a hyperbolic geodesic in the complement of
the remaining arcs $\gamma\smallsetminus \gamma_j$. These curves are
characterized by the property that their conformal welding is piecewise
M\"obius. Among other things, we compute the Schwarzian derivatives of the
Riemann maps of the two regions in $\hat {\mathbb C}\smallsetminus \gamma$,
show that they form a rational function with second order poles at the
endpoints of the $\gamma_j,$ and show that the poles are simple if the curve
has continuous tangents. Our key tool is the explicit computation of all
geodesic pairs, namely pairs of chords $\gamma=\gamma_1\cup\gamma_2$ in a
simply connected domain $D$ such that $\gamma_j$ is a hyperbolic geodesic in
$D\smallsetminus \gamma_{3-j}$ for both $j=1$ and $j=2$.
