We study arrangements of geodesic arcs on a sphere, where all arcs are internally disjoint and each arc has its endpoints located within the interior of other arcs. We establish fundamental results concerning the minimum number of arcs in such arrangements, depending on local geometric constraints such as "one-sidedness" and "k-orientation". En route to these results, we generalize and settle an open problem from CCCG 2022. Namely, we prove that any such arrangement has at least two "clockwise swirls" and at least two "counterclockwise swirls".

翻译：我们研究球面上测地弧的排列，其中所有弧内部不相交，且每条弧的端点位于其他弧的内部。我们建立了此类排列中弧的最小数量的基本结果，该结果依赖于局部几何约束，如“单侧性”和“k-定向”。在得出这些结果的过程中，我们推广并解决了CCCG 2022中的一个开放问题。具体而言，我们证明了任何此类排列至少有两个“顺时针旋涡”和至少两个“逆时针旋涡”。