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

Piecewise geodesic Jordan curves I: weldings, explicit computations, and Schwarzian derivatives