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".