## 3 Analysis
### Convergence Analysis
#### 3.1.2 Convergence in Two-Dimensional Setup  
Similar approach can be adopted for the illustration of convergence property in a two-dimensional setup. Consider \(n\) beacons where \(n\geq 2\). Each beacon is located at coordinate \(s_{i},i=1,2,...,n\). All distance variables are normalized to the length of the spring, i.e. the length of the spring is of one unit long. We shall arbitrarily draw a straight line crossing the only stopping point, then rotate the line along with all beacons such that the line becomes horizontal and overlaps with an x-axis, and finally shift the line horizontally such that the stopping point stays at \(\hat{x}\) on the x-axis. After this transformation, let the xy-coordinate of the \(i\)-th beacon be \((x_{i},y_{i})\), \(i=1,2,...,n\). We now focus on the image of the forces projected on the straight line, which can be determined by  
\[F_{x_{i}}(x)=(x-x_{i})\left(\frac{1}{\sqrt{(x-x_{i})^{2}+y_{i}^{2}}}-1\right). \tag{5}\]  
We call the zone where a particular spring experiences compression a compression zone of the spring or the beacon. If the straight line does not cross the compression zone of a beacon, then we have \(|y_{i}|>1\) which ensures that the spring remains stretched along the line. It can be shown that with \(|y_{i}|>1\), \(F_{x_{i}}(x)\) given in (5) is a monotone decreasing continuous function. We plot the result of (5) with \(x_{i}=1\) and \(y_{i}=2\) in Figure 2 to illustrate its monotonic decreasing characteristics.  
Figure 2: The plot of \(F_{x_{i}}(x)\) for Equations (4) and (5) with \(x_{i}=1\) and different \(y_{i}\) values.  
For the case that the straight line touches or crosses the compression zone of a beacon, we have \(\left|y_{i}\right|\leq 1\). It can be shown that \(F_{x_{i}}(x)\) given in (5) decreases monotonically except in the range \(\Big{(}-\sqrt{-y_{i}^{2}+y_{i}^{\frac{4}{3}}}\), \(\sqrt{-y_{i}^{2}+y_{i}^{\frac{4}{3}}}\Big{)}\) where it increases monotonically for no more than two units in total. In fact, (4) in the 1-D setup is a special case of (5) where the straight line also crosses the beacon. In Figure 2, the cases of \(\left|y_{i}\right|\leq 1\) and \(y_{i}=0\) both with \(x_{i}=1\) are also plotted illustrating a similar characteristics between the case \(\left|y_{i}\right|\leq 1\) in the 2-D setup and the case \(y_{i}=0\) in the 1-D setup.  
Using this setup, following the proof given in Proposition 1, it can be shown that with the existence of a unique stopping point, at any other location along the straight line, there exist an net force where its image projected on the straight line points towards the stopping point which moves the node nearer to the stopping point. As the algorithm iterates, the location of the node will eventually converged to the stopping point. Note that since we only show that the image of a force along the line connecting the stopping point and the arbitrary chosen location points towards the the stopping point, the node may not necessarily moves along the straight line back to the only stopping point. The node may move to a new location nearer to the stopping point not necessarily on the straight line, but we can redraw another line for that new location and then apply the same argument to show that the next location of the node will again be nearer to the stopping point than that of the current location.