A numerical approach for the filtered generalized Cech complex

TL;DR

This paper introduces a numerical algorithm to compute the filtered generalized Cech complex for disks of varying radii in the plane, with applications to minimal enclosing ball problems.

Contribution

It presents a novel numerical method for calculating the filtered generalized Cech complex and extends the approach to higher-dimensional disks, supported by a generalized Vietoris-Rips Lemma.

Findings

Algorithm guarantees convergence for the scale factor calculation.

Provides a method to compute the minimal enclosing ball for point sets.

Extends the computation to higher-dimensional disks in Euclidean space.

Abstract

In this paper, we present an algorithm to compute the filtered generalized \v{C}ech complex for a finite collection of disks in the plane, which don't necessarily have the same radius. The key step behind the algorithm is to calculate the minimum scale factor needed to ensure rescaled disks have a nonempty intersection, through a numerical approach, whose convergence is guaranteed by a generalization of the well-known Vietoris-Rips Lemma, which we also prove in an alternative way, using elementary geometric arguments. We present two applications of our main results. We give an algorithm for computing the 2-dimensional filtered generalized \v{C}ech complex of a finite collection of $d$-dimensional disks in $\mathbb{R}^d$. In addition, we show how the algorithm yields the minimal enclosing ball for a finite set of points in the plane.