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, proving that any such arrangement has at least two "clockwise swirls" and at least two "counterclockwise swirls".

翻译：我们研究了球面上测地弧的排列，其中所有弧内部互不相交，且每条弧的端点位于其他弧的内部。根据“单侧性”和“k-定向”等局部几何约束，我们建立了此类排列中弧数最小值的基本结果。在得出这些结果的过程中，我们推广并解决了CCCG 2022中的一个未解决问题，证明任意此类排列至少包含两个“顺时针旋涡”和两个“逆时针旋涡”。