# A Numerical Approach for the Filtered Generalized Cech Complex
## 2 Vietoris-Rips and Cech systems  
Throughout this paper, a finite collection of closed \(d\)-disks in the euclidean space \(\mathbb{R}^{d}\), with positive radius,  
\[M=\{D_{i}(c_{i};r_{i})\subset\mathbb{R}^{d}\mid r_{i}>0,1\leq i\leq m\} \tag{1}\]  
will be called \(d\)-disk system, or simply disk system when there is no risk of confusion. In this section, we introduce and analyze two fundamental subclasses of disk systems, namely, the Vietoris-Rips systems and the Cech systems. We study the infimum of those scales that turn a disk system into a Vietoris-Rips or Cech system. We conclude this section presenting a generalized version of the Vietoris-Rips Lemma, extended to disk systems. Definition 1: Let \(M=\{D_{1},D_{2},\ldots,D_{m}\}\) be a disk system. We say \(M\) is a Vietoris-Rips system if \(D_{i}\cap D_{j}\
eq\emptyset\) for each pair \(i,j\in\{1,2,\ldots,m\}\). Moreover, if the disk system \(M\) has the nonempty intersection property \(\bigcap_{D_{i}\in M}D_{i}\
eq\emptyset\), then \(M\) is called a Cech system. For each \(\lambda\geq 0\), and disk system \(M\) as in (1) we define the collection  
\[M_{\lambda}:=\{D_{i}(c_{i};\lambda r_{i})\subset\mathbb{R}^{d}\mid D_{i}\in M\}\]and say that \(\lambda\) is a scale. Geometrically, the set \(M_{\lambda}\) consists of disks with the same centers than those in \(M\), but with rescaled radii by \(\lambda\). Clearly, only when \(\lambda>0\) the set \(M_{\lambda}\) will be again a disk system. Note that \(M_{1}=M\), and \(M_{0}\) is the set consisting of the centers of the disks in \(M\). Definition 2: Let \(M\) be a disk system. The Vietoris-Rips scale of \(M\) is defined by,  
\[\
u_{M}:=\inf\{\lambda\in\mathbb{R}\mid M_{\lambda}\text{ is a Vietoris-Rips system}\}.\]  
Analogously, the \(\hat{\text{C}}\)ech scale of \(M\) is defined by,  
\[\mu_{M}:=\inf\{\lambda\in\mathbb{R}\mid M_{\lambda}\text{ is a $\hat{\text{C}}$ech system}\}.\]  
Let \(\mu_{M}\) be the \(\hat{\text{C}}\)ech scale of the disk system \(M\), then we have that \(\bigcap_{D_{i}\in M}D_{i}(c_{i};\mu_{M}r_{i})\
eq\emptyset\). Essential, this is a consequence of the completeness of the euclidean space, the fact that \(\bigcap_{D_{i}\in M}D_{i}(c_{i};\lambda r_{i})\subset\mathbb{R}^{d}\) is a closed subset for every scale \(\lambda\) in the set \(\{\lambda\in\mathbb{R}\mid M_{\lambda}\text{ is a $\hat{\text{C}}$ech system}\}\) and \(\bigcap_{D_{i}\in M}D_{i}(c_{i};\lambda^{\prime}r_{i})\subset\bigcap_{D_{i} \in M}D_{i}(c_{i};\lambda r_{i})\) for \(\lambda^{\prime}<\lambda\). A straightforward calculation shows the following characterizations. \(M\) is a Vietoris-Rips system if and only if, \(\
u_{M}\leq 1\) (in particular, \(\
u_{M_{\
u_{M}}}=1\)); similarly, \(M\) is a \(\hat{\text{C}}\)ech system if and only if, \(\mu_{M}\leq 1\). Note that it is easy to calculate the Vietoris-Rips scale \(\
u_{M}\) for a given disk system \(M=\{D_{1},D_{2},\ldots,D_{m}\}\): if \(r_{i}\) denotes the radius of \(D_{i}\), and \(\|c_{i}-c_{j}\|\) represents the distance between the center of \(D_{i}\) and \(D_{j}\), then \(\
u_{M}=\max_{i<j}\{\|c_{i}-c_{j}\|/(r_{i}+r_{j})\}\). For a disk system \(M\) with just one disk, its Vietoris-Rips scale is \(\
u_{M}=0\); if \(M=\{D_{1},D_{2}\}\) has two disks, then \(\
u_{M}=\|c_{1}-c_{2}\|/(r_{1}+r_{2})\). Actually, in both cases the Vietoris-Rips scale agrees with the \(\hat{\text{C}}\)ech scale. On the other hand, calculating the \(\hat{\text{C}}\)ech scale is a more complicated issue if the disk system has at least three disks. Concerning to \(\hat{\text{C}}\)ech scales, we have the following lemma, which will become important for our implementations. Lemma 1: _Let \(\mu\geq 0\) be a scale and let \(M\) be a disk system. Then, \(\mu\) is the \(\hat{\text{C}}\)ech scale of \(M\) if and only if, the \(\mu\)-rescaled system \(M_{\mu}\) has only one intersection point, i  
**Example 1**.