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 $(\widehat{\mathbb C} \smallsetminus \gamma)\cup \gamma_j$. 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 $\widehat{\mathbb C} \smallsetminus \gamma$ 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 $D\smallsetminus \gamma_{3-j}$ for both $j=1$ and $j=2$.