# A Numerical Approach for the Filtered Generalized Cech Complex
## 1 Introduction  
Recently, in the study of data point clouds from a topological approach (cf. [1; 2; 3; 4; 5]), the need to develop algorithms to construct different simplicial structures has arisen, such as the Vietoris-Rips complex, the Cech complex, the piecewise linear lower star complex, etc. (cf. [6; 7]). Of particular interest to us is the generalized Cech complex structure, whereas the standard Cech complex is induced by the intersection of a collection of disks with fixed radius, the generalized version admits different radii (see [8]); when radii are rescaled, using the same scale factor each time, the corresponding simplicial complexes forms the filtered generalized Cech complex. There exist efficient algorithms to calculate the standard Cech complex (see, e.g., in [9]), and software currently available to obtain the associated filtration (cf. [10; 11]); also, in [12] the authors propose an algorithm to approximate the Cech filtration. On the other hand, we can find algorithms to calculate the generalized Cech complex (see, e.g., in [8]); however, as far as we know, there are neither algorithms nor software to provide the filtered generalized Cech complex. In the present work, we show an algorithm to compute the filtered generalized Cech complex for a finite collection of disks, specifically, in the plane. Actually, we also show an algorithm to build up to the 2-dimensional filtered generalized structure (or 2-skeleton), for higher-dimensional disk systems, which many applications only require, as we can see in [13; 14; 15; 16; 17]. The key step behind these proposed algorithms, is to calculate the minimum scale factor (called Cech scale) needed to ensure that the rescaled disks have a nonempty intersection; the generalized Vietoris-Rips lemma over multiple radii will allow us to calculate these scales numerically. We must emphasize that our main algorithm (Algorithm 3) is only generalizable to higher-dimensional disk systems to obtain the 2-dimensional filtered generalized Cech structure, as we show as an application. Additionally, we show how our algorithm yields the minimal enclosing ball for a finite set of points in the plane. This paper is organized as follows. In Section 2, we introduce basic notions and notation which will be used throughout the paper. We define the Vietoris-Rips system and the Cech system, associated to a finite collection of closed disks in the euclidean space (or disk system) in terms of their intersection. We also introduce the fundamental notions of Vietoris-Rips scale and Cech scale for a disk system, as the infimum over all rescaling factors such that the disk system becomes a Vietoris-Rips system or a Cech system, respectively. In Lemma 2, we state and prove a generalization, over multiple radii, of the well-known Vietoris-Rips Lemma [17] (Theorem 2.5) using elementary geometric arguments. In [18] there is a proof in the generalized case, following the ideas in [17]. In Section 3 we describe the generalized versions of standard Vietoris-Rips and Cech simplicial complex structures, to the case of disk systems with different radii. We explain how their respective filtrations are induced by weight functions, and we propose an algorithm to obtain the Cech-weight function of a given disk system, associating to each Cech simplex its corresponding Cech scale. Section 4 focuses on studying the intersection properties of collections of disks in the plane. We define a real-valuated function associated to each disk system in the plane, such that, if it turns out to be non-negative, then its Cech scale agrees with its Vietoris-Rips scale, being then easy to compute; otherwise, the Cech scale will correspond to a root of such function, and we propose a numerical approach to obtain this Cech scale (Section 5), supported on the generalized Vietoris-Rips Lemma which provides appropriated bounds.