# A Numerical Approach for the Filtered Generalized Cech Complex
## 4 Intersection Properties of Disk Systems
Definition 3: Let \(M=\{D_{1},D_{2},\ldots,D_{m}\}\) be a Vietoris-Rips system in the plane, with \(m\geq 3\). We define  
\[\rho(M):=\max_{1\leq i,j\leq m}\left\{\min_{k\
eq i,j}\{r_{k}-\|d_{ij}-c_{k}\| \}\right\}.\]  
If \(\
u_{M}\) is the Vietoris-Rips scale of \(M\), then we define the map \(\rho_{M}:[\
u_{M},\infty)\to\mathbb{R}\), \(\lambda\mapsto\rho_{M}(\lambda)=\rho(M_{\lambda})\). Given three disks \(D_{i}(c_{i};r_{i})\), \(D_{j}(c_{j};r_{j})\) and \(D_{k}(c_{k};r_{k})\) in the 2-disk system \(M=\{D_{1},\ldots,D_{m}\}\), with Vietoris-Rips scale \(\
u_{M}\), denote by \(\Lambda^{k}_{i,j}:[\
u_{M},\infty)\to\mathbb{R}\) the map \(\lambda\mapsto\lambda r_{k}-\|d_{ij}(\lambda)-c_{k}\|\), where \(d_{ij}(\lambda)\) is the element in \(D_{i}(c_{i};\lambda r_{i})\cap D_{j}(c_{j};\lambda r_{j})\). In other words, \(\Lambda^{k}_{i,j}(\lambda)\) is the signed distance from the point \(d_{ij}(\lambda)\) to the set \(D_{k}(c_{k};\lambda r_{k})\). If \(r_{i}\
eq r_{j}\), then for each \(k\
eq i,j\) the map \(\lambda\mapsto\lambda r_{k}-\|d_{ij}(\lambda)-c_{k}\|\) is defined and continuous in the closed interval \(\big{[}\|c_{i}-c_{j}\|/(r_{i}+r_{j}),\|c_{i}-c_{j}\|/|r_{i}-r_{j}|\big{]}\), as it is the signed distance from an intersection point of two continuously deforming curves (therefore its position vary continuously as long as the intersection exists) to the continuously deforming set \(D_{k}(c_{k};\lambda r_{k})\) with respect to \(\lambda\). Also, the map \(\lambda\mapsto\lambda r_{k}-\|d_{ij}(\lambda)-c_{k}\|\) vary linearly in the range \(\big{[}\|c_{i}-c_{j}\|/|r_{i}-r_{j}|,\infty)\) because for \(\lambda\geq\|c_{i}-c_{j}\|/|r_{i}-r_{j}|\), the term \(\|d_{ij}(\lambda)-c_{k}\|\) remains constant. The left picture in Figure 4 shows in bold red color the geometric place of \(\{d_{ij}(\lambda),d_{ji}(\lambda)\}\), which vary continuously respect to the parameter \(\lambda\) and also the distance from it to the fix point \(c_{k}\). On the other hand, for \(r_{i}=r_{j}\), the points \(\{d_{ij}(\lambda),d_{ji}(\lambda)\}\) vary continuously respect to \(\lambda\) on the line showed in the right picture of Figure 4. Therefore, \(\lambda r_{k}-\|d_{ij}(\lambda)-c_{k}\|\) also depend continuously of \(\lambda\). From the above argument, each map \(\Lambda^{k}_{i,j}\) is continuous in the interval \([\
u_{M},\infty)\), and by the continuity of the min-max functions and that  
\[\rho_{M}(\lambda)=\max_{1\leq i,j\leq m}\left\{\min_{k\
eq i,j}\Lambda^{k}_{i,j}(\lambda)\right\},\]  
Figure 4: Geometric place of \(d_{ij}(\lambda)\). it follows that \(\rho_{M}\) is also a continuous map in the interval \([\
u_{M},\infty)\). However, the map \(\rho_{M}\) is not differentiable, in general, to every point in such interval. The map \(\rho_{M}\) plays a key role in the rest of this work. We present the next characterization of Cech systems in terms of \(\rho_{M}\). Lemma 4: _Let \(M\) be a 2-disk system. Then \(M_{\lambda}\) is a Cech system if and only if, \(\rho_{M}(\lambda)\geq 0\). In particular \(\rho_{M}(\sqrt{4/3}\
u_{M})\geq 0\)._  
Proof: By Lemma 3, \(M_{\lambda}\) is a Cech system in the plane if and only if, there exists \(d_{ij}(\lambda)\) such that \(d_{ij}(\lambda)\in D_{k}(c_{k};\lambda\,r_{k})\) for every \(k\
eq i,j\), i.e., \(\Lambda^{k}_{i,j}(\lambda)\geq 0\) for every \(k\
eq i,j\), which is equivalent to \(\rho_{M}(\lambda)\geq 0\). On the other hand, from Corollary 1 and taking \(d=2\), the rescaled system \(M_{\sqrt{4/3}\
u_{M}}\) is a Cech system, then by the first assertion, \(\rho_{M}(\sqrt{4/3}\
u_{M})\geq 0\).