# A Numerical Approach for the Filtered Generalized Cech Complex
## 5 The Cech.scale Algorithm
It is possible, for some configurations of the 2-disk system, that the map \(\rho_{M}\) has a behavior as in Figure 5.  
The last iterative part is a finite process because \(\rho_{M}\) is algebraic over \(\mathbb{Q}\), then eventually the set \(\sqcap M_{\mu^{*}}\) will be unitary and the Cech scale will be calculated. The step (8) in Algorithm 2 is necessary, as show the following example, in which the map \(\rho_{M}\) has another root along side the Cech scale in the interval \([v_{M},\sqrt{4/3}v_{M}]\). **Example 3**. : _Let \(M=\{D_{1},D_{2},D_{3}\}\) be the 2-disk system in Figure 6. A direct calculation, yields that \(v_{M}=\mu_{M}=0.8947\). On the other hand, we also have that \(\rho_{M}(v_{M})=\rho_{M}(\mu_{M})=\rho_{M}(\lambda)=0\) for \(\lambda=1\). Therefore, the map \(\rho_{M}\) has more than one root on the interval \([v_{M},\sqrt{4/3}v_{M}]=[0.8947,1.0331]\)._  
In Example 3, the Vietoris-Rips scale \(v_{M}\), of the 2-disk system \(M\), agrees with the Cech scale \(\mu_{M}\); however is possible to construct more sophisticated (and symmetric) disk system \(M\) such that \(v_{M}<\mu_{M}\) and \(\rho_{M}(v_{M})<0<\rho_{M}(\mu_{M})\), for which there exists \(\mu^{\prime}\in(v,\sqrt{4/3}v_{M})\) with \(\rho_{M}(\mu^{\prime})=0\). On the other hand, if the 2-disk system \(M\) consists of just three disks and \(\rho_{M}(v_{M})<0\), then its Cech scale can be computed with only one application of the numerical method, as we asseverate in the following lemma. **Lemma 6**. : _Let \(M=\{D_{1},D_{2},D_{3}\}\) be a 2-disk system such that \(\rho_{M}(v_{M})<0\). Then, there exists a unique root of the map \(\rho_{M}\) in \([v_{M},\sqrt{4/3}v_{M}]\). Thus, \(\mu_{M}\) will be the output of \(\mathsf{bisection}(\rho_{M},v_{M},\sqrt{4/3}v_{M})\)._  
Proof. : It is straightforward to verify that \(\rho_{M}(v_{M})\geq 0\) for any configuration with \(c_{1}\), \(c_{2}\) and \(c_{3}\) collinear. Thus, it follows \(\{c_{1},c_{2},c_{3}\}\) is in general position. Let \(\mu_{M}\) be the Cech scale of the 2-disk system \(M\) and \(c_{M}\) the intersection point. Define \(A(\lambda):=\ \cap_{i=1}^{3}D_{i}(c_{i};\lambda r_{i})\). Note that \(A(\mu_{M})=\{c_{M}\}\). We claim that there exist at least two distinct points \(p\) and \(q\), in the set  
\[\{d_{ij}(\lambda)\in D_{i}(c_{i};\lambda r_{i})\cap D_{j}(c_{j};\lambda r_{j} )\mid D_{i},D_{j}\in M_{\lambda}\}\subset\partial A(\lambda).\]  
This is evident if \(\partial A(\lambda)\) is given by two or more circumference arcs. On the other hand, if \(\partial A(\lambda)=\partial D_{i}(c_{i};\lambda r_{i})\) for some \(1\leq i\leq 3\), then \(d_{ij}(\lambda),d_{ik}(\lambda)\in D_{i}(c_{i};\lambda r_{i})\) for \(j\
eq i\) and \(k\
eq i\). Moreover, \(d_{ij}(\lambda)\
eq d_{ik}(\lambda)\) since \(\{c_{1},c_{2},c_{3}\}\) is not a collinear set. If both points \(p\) and \(q\) belongs to each boundary of the three disks, \(p,q\in\cap_{i=1}^{3}\partial D_{i}(c_{i};\lambda r_{i})\), then \(\{c_{1},c_{2},c_{3}\}\) would be also a collinear set. Without loss of generality, we suppose that \(p=d_{23}(\lambda)\
ot\in\partial D_{1}(c_{1};\lambda r_{1})\). Then, \(\rho_{M}(\lambda)\geq\lambda r_{1}-\|d_{23}(\lambda)-c_{1}\|>0\), and the lemma follows. Of course, the choose of the indexes depend of the value of \(\lambda\), but the above arguments show that always there exist such combination which guarantee that \(\rho_{M}(\lambda)\) is positive for \(\lambda>\mu_{M}\). Figure 6: The 2-disk system \(M\). The following algorithm takes advantage of the unicity property for the root of \(\rho_{M}\), in a 2-disk system with three disks. Essentially, the algorithm consist in iterating the Algorithm 2 systematically over every triplet of disks from \(M\). ```
Input :A 2-disk system \(M\). Output :Cech scale \(\mu_{M}\) and intersection point \(c_{M}\).