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

We consider Jordan curves of the form $\gamma ={\cup }_{j=1}^n{\gamma }_j$ on the Riemann sphere for which each ${\gamma }_j$ is a hyperbolic geodesic in $(\widehat{\mathbb{C}}\setminus \gamma )\cup {\gamma }_j$. These Jordan curves are characterized by their conformal welding being piecewise M"obius. We show that the Schwarzian derivatives of the uniformizing mappings of the two regions in $\widehat{\mathbb{C}}\setminus \gamma$ form a rational function with at most second-order poles at the endpoints of ${\gamma }_j$ and that the poles are simple if the curve has continuous tangents. A key tool is the explicit computation of all $C^1$ geodesic pairs, namely $C^1$ chords $\gamma ={\gamma }_1\cup {\gamma }_2$ in a simply connected domain $D$ such that ${\gamma }_j$ is a hyperbolic geodesic in $D\setminus {\gamma }_{3-j}$ for both $j=1$ and $j=2$.