# A Numerical Approach for the Filtered Generalized Cech Complex
## 4 Intersection Properties of Disk Systems  
In this section, we focus on studying disk systems in the plane, i.e., 2-disk systems. As we have seen in the last section, the study of the Cech scale is a key aspect to the construction and study of filtered generalized Cech complex. In this section, we establish several intersection properties of 2-disk systems, which will lead us to be able to calculate the Cech scale. Let \(\partial D_{i}(c_{i};r_{i}):=\{x\in\mathbb{R}^{2}\mid\|x-c_{i}\|=r_{i}\}\) be the boundary of the closed 2-dimensional disk \(D_{i}(c_{i},r_{i})\subset\mathbb{R}^{2}\). Let \(D_{i}\) and \(D_{j}\) be two closed disks in the plane, such that \(D_{i}\cap D_{j}\
eq\emptyset\). We define \(D_{i}\cap D_{j}\) to be the unitary set \(\{d_{ij}\}\) constructed as follows. 1. If \(\partial D_{i}\cap\partial D_{j}\
eq\emptyset\), then \(d_{ij}\in\partial D_{i}\cap\partial D_{j}\) is the only one point with the property \(\langle d_{ij}-c_{i},\mathbf{n}_{ij}\rangle\geq 0\), where \(\mathbf{n}_{ij}=(-b,a)\) is the normal vector to \(c_{j}-c_{i}=(a,b)\),
2. If \(\partial D_{i}\cap\partial D_{j}=\emptyset\), we define \(d_{ij}\) as the unique intersection point in \(\partial D_{i}(c_{i};\lambda r_{i})\cap\partial D_{j}(c_{j};\lambda r_{j})\), for \(\lambda\) given as the minimal scale such that \(D_{i}(c_{i};\lambda r_{i})\subset D_{j}(c_{j};\lambda r_{j})\) or \(D_{j}(c_{j};\lambda r_{j})\subset D_{i}(c_{i};\lambda r_{i})\), i.e., \(\lambda=\|c_{i}-c_{j}\|/|r_{i}-r_{j}|\). Clearly, if \(\partial D_{i}\cap\partial D_{j}=\emptyset\), then \(d_{ij}=d_{ji}\). In particular, when \(D_{i}\) and \(D_{j}\) are concentric, then \(D_{i}\cap D_{j}=\{c_{i}\}=\{c_{j}\}\). On the other hand, if the closed disks \(D_{i}\) and \(D_{j}\) are internally or externally tangent, then \(d_{ij}=d_{ji}\). We can think about \(d_{ij}\), when \(\partial D_{i}\cap\partial D_{j}\) is not empty, as the intersection point of the boundaries at the left of the vector from \(c_{i}\) to \(c_{j}\). Figure 3 shows the above construction. We will denote by \(d_{ij}(\lambda)\), instead of simply \(d_{ij}\), for the intersection point of the \(\lambda\)-rescaled disks \(D_{i}(c_{i};\lambda r_{i})\) and \(D_{j}(c_{j};\lambda r_{j})\). In order to study Cech systems, we give the following characterization, according the intersection points \(d_{ij}\). **Lemma 3**: _Let \(M=\{D_{1},D_{2},\ldots,D_{m}\}\) be a 2-disk system. Then \(M\) is a Cech system if and only if, there exist \(D_{i},D_{j}\in M\) such that \(d_{ij}\in D_{i}\cap D_{j}\) satisfies \(d_{ij}\in D_{k}\) for all \(1\leq k\leq m\)._  
**Proof. ** Suppose \(M\) is a Cech system. Define \(A:=\bigcap_{1\leq i\leq m}D_{i}\
eq\emptyset\); then, \(A\) has only one of the following geometries. (i) \(A=\{c_{M}\}\),  
(ii) \(A\) is a region bounded by more than one circumference arc,  
(iii) \(A=D_{i_{0}}\) for some \(i_{0}\in\{1,\ldots,m\}\). In the first case, necessarily \(c_{M}\) belongs to the boundary of two or more disks. Let \(D_{i}\) and \(D_{j}\) be two disks in \(M\) such that \(c_{M}\in\partial D_{i}\cap\partial D_{j}\), it follows that \(c_{M}=d_{ij}\) or \(c_{M}=d_{ji}\), in both cases the lemma holds. Figure 3: Intersection point \(d_{ij}\). For the second case, if \(a\in\partial A\subset A\) belongs to the boundary and is in the intersection of two arcs, say \(\partial D_{i}\) and \(\partial D_{j}\), then \(a=d_{ij}\) or \(a=d_{ji}\), and it satisfy \(a\in D_{k}\) for every \(1\leq k\leq m\). For the last case, if \(A=D_{i_{0}}\) for some \(i_{0}\), then for each \(j\
eq i_{0}\) we have \(d_{i_{0}j}\in D_{i_{0}}=A\) and all of these points belongs to \(D_{k}\) for all \(D_{k}\in M\). Therefore, in any case, there exists such point \(d_{ij}\). The converse is clear by definition of a Cech system. This criterion was presented in ([8], Section III) for a 2-disk system. Next, we define the map \(\rho\), a key tool for the rest of this work. This map will allow us to discern the minimal scale in which a 2-disk system has the nonempty intersection property.