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Intersection properties of finite disk collections

Jesús F. Espinoza, Cynthia G. Esquer-Pérez

TL;DR

This work analyzes intersection properties of finite disk collections in $\\mathbb{R}^d$ and develops practical tools for generalized Čech complexes by leveraging pole-based tests and Helly-type reductions. It derives explicit center–radius formulas for the intersection sphere of two disks and extends to $m$ disks, providing a framework to compute poles and test nonempty intersections efficiently. The authors introduce algorithms for recognizing Čech systems and computing the Čech scale via bisection, and integrate a minimal axis-aligned bounding box (AABB) construction with a Helly-based reduction to handle higher-order disk intersections. The resulting methodologies enable robust, dimension-agnostic analysis of disk intersections, with applications to topology-driven data analysis and collision/spatial queries in higher dimensions.

Abstract

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 Čech scale, and constructing the minimal Axis-Aligned Bounding Box (AABB) that encloses the intersection of all disks in the system.

Intersection properties of finite disk collections

TL;DR

This work analyzes intersection properties of finite disk collections in and develops practical tools for generalized Čech complexes by leveraging pole-based tests and Helly-type reductions. It derives explicit center–radius formulas for the intersection sphere of two disks and extends to disks, providing a framework to compute poles and test nonempty intersections efficiently. The authors introduce algorithms for recognizing Čech systems and computing the Čech scale via bisection, and integrate a minimal axis-aligned bounding box (AABB) construction with a Helly-based reduction to handle higher-order disk intersections. The resulting methodologies enable robust, dimension-agnostic analysis of disk intersections, with applications to topology-driven data analysis and collision/spatial queries in higher dimensions.

Abstract

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 Čech scale, and constructing the minimal Axis-Aligned Bounding Box (AABB) that encloses the intersection of all disks in the system.
Paper Structure (12 sections, 9 theorems, 55 equations, 5 figures, 3 algorithms)

This paper contains 12 sections, 9 theorems, 55 equations, 5 figures, 3 algorithms.

Table of Contents

  1. Introduction
  2. Intersection properties of sphere systems
  3. Sphere systems with two spheres
  4. Sphere systems with more than two spheres
  5. Vietoris-Rips and Cech systems
  6. Algorithm for determining Čech system.
  7. Algorithm to compute the Cech scale.
  8. Minimal Axis-Aligned Bounding Box.
  9. Minimal axis-aligned bounding box for two disks
  10. Minimal axis-aligned bounding box for more than two disks
  11. Minimal axis-aligned bounding box Algorithm.
  12. Acknowledgements

Key Result

Lemma 2

Let $M = \{D_1(c_1; r_1), \ldots, D_m(c_m; r_m)\}$ be disk system such that $\{c_1, \dots, c_m\}$ is a set in general position in $\mathbb{R}^d$. Then, the possibilities for the set $\cap_{D_i \in M} \partial D_i$ are:

Figures (5)

  • Figure 1: Visualization of the subspaces $L$ and $M+\{c\}$.
  • Figure 2: AABB of two disks.
  • Figure 3: Disk system $M$ projected onto the $xy$-plane.
  • Figure 4: Disk system $N$ projected onto the $xy$-plane.
  • Figure 5:

Theorems & Definitions (22)

  • Definition 1
  • Lemma 2
  • Definition 3
  • Lemma 4
  • proof
  • Lemma 5
  • proof
  • Lemma 6
  • proof
  • Definition 7
  • ...and 12 more