In this work we study the intersection properties of a finite disk system in the euclidean space. We accomplish this by utilizing subsets of spheres with varying dimensions and analyze specific points within them, referred to as poles. Additionally, we introduce two applications: estimating the common scale factor for the radii that makes the re-scaled disks intersects in a single point, this is the \v{C}ech scale, and constructing the minimal Axis-Aligned Bounding Box (AABB) that encloses the intersection of all disks in the system.

翻译：本文研究了欧几里得空间中有限圆盘系统的交集性质。我们通过利用不同维度的球面子集并分析其中的特定点（称为极点）来实现这一目标。此外，我们介绍了两个应用：估计使重新缩放后的圆盘交于一点的公共半径比例因子（即Čech尺度），以及构造包围系统中所有圆盘交集的最小轴对齐包围盒。