# Self-Organization When Pedestrians Move in Opposite Directions. Multi-Lane Circular Track Model
## 4 Possible Collision Times  
We recall the reader that two individuals collide at time \(t\) if they are in the same lane at that time and their angles differ by an integer multiple of \(2\pi\). When we say they are in the same lane at time \(t\), we mean they are in the same lane just before one of them changes lanes. Note that, since \(0\) is an integer, the angles differing by an integer multiple of \(2\pi\) includes the case when the angles are equal. We define \(\tau_{ij}\) to be the first time that the angle of the individual \(i\) moving counterclockwise and the angle of the individual \(j\) moving clockwise differ by an integer multiple of \(2\pi\). These two individuals do not collide for any \(t\) satisfying \(0<t<\tau_{ij}\) and they collide at \(t=\tau_{ij}\) if and only if they are in the same lane at that time. Thus, we refer to \(\tau_{ij}\) as the first or smallest possible collision time between the individual \(i\) moving counterclockwise and the individual \(j\) moving clockwise. Recall that the angle of the individual \(i\) moving counterclockwise and the angle of the individual \(j\) moving clockwise are given by \(\theta_{i}^{(+)}(t)=\theta_{i}^{(+)}(0)+\omega t\) and \(\theta_{j}^{(-)}(t)=\theta_{j}^{(-)}(0)-\omega t\), respectively. Recall also that all the initial angles are greater than or equal to \(0\) and less than \(2\pi\). Given these two facts, little thought is necessary to convince the reader about the validity of the Equation (1). Note that the time \(\tau_{ij}\) always satisfies \(0<\tau_{ij}<\pi/\omega\). Figure 1: Positions of the pedestrians at different times. The circles are the pedestrians moving clockwise. The axes are the pedestrians moving counterclockwise. We define \(\mathcal{T}_{ij}\) to be the set of all positive times when the difference between the angles of the individual \(i\) moving counterclockwise and the individual \(j\) moving clockwise differ by an integer multiple of \(2\pi\). Straight forward algebra shows that \(\mathcal{T}_{ij}\) is the set of all times of the form \(\tau_{ij}+n\pi/\omega\), where \(n\) is any non-negative integer,  
\[\mathcal{T}_{ij}=\left\{t=\tau_{ij}+n\frac{\pi}{\omega}\text{ for some non-negative integer }n\right\}. \tag{2}\]  
If the individual \(i\) moving counterclockwise and the individual \(j\) moving clockwise collide, they so do at some time \(t\) in \(\mathcal{T}_{ij}\). In fact, these two individuals collide at time \(t\) if and only if \(t\in\mathcal{T}_{ij}\) and they are both in the same lane at time \(t\). Given these facts, we refer to the set \(\mathcal{T}_{ij}\) as the set of possible collision times between the individual \(i\) moving counterclockwise and the individual \(j\) moving clockwise. We define \(\mathcal{T}\) to be the union of the sets \(\mathcal{T}_{ij}\), where the union is taken over all individuals \(i\) moving counterclockwise and individuals \(j\) moving clockwise  
\[\mathcal{T}=\bigcup_{1\leq i,j\leq N}\mathcal{T}_{ij}. \tag{3}\]  
If there is a collision at time \(t\), then \(t\) is in the set \(\mathcal{T}\). Thus, we refer to \(\mathcal{T}\) as the set of possible collision times. We sort the elements in \(\mathcal{T}\) and label them as \(t_{n}\) with \(n\geq 1\). In other words, the sequence \(t_{n}\), with \(n\geq 1\), is defined by the facts that \(t_{n}<t_{n+1}\) for all positive integers n, and  
\[\mathcal{T}=\left\{t_{n}:n\geq 1^{\prime}\right\}. \tag{4}\]  
For convenience, we define \(t_{0}=0\), but we do keep \(t_{0}\) out of the set \(\mathcal{T}\). Note that the assumption in Point 9 in Section 2 implies that the sets \(\mathcal{T}_{ij}\) are pairwise disjoint. Thus, \(t_{n}\) is a first possible collision time between two individuals, i.e., \(t_{n}=\tau_{ij}\) for some \(i\) and \(j\), if and only if \(1\leq n\leq N^{2}\). Thus, we have that \(t_{N^{2}}<\pi/\omega<t_{N^{2}+1}\). Note also that \(t_{N^{2}+n}=t_{n}+\pi/\omega\) for all \(n\geq 1\).