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，其中所有弧段内部互不相交，且每条弧的端点位于其他弧段的内部。我们根据局部几何约束（如“单侧性”和“k定向”）建立了这类排列中弧段最小数量的基本结果。在获得这些结果的过程中，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包含两个“顺时针漩涡”和两个“逆时针漩涡”。