# Tracking Mobile Sinks via Analysis of Movement Angle Changes in WSNs
## 3 Details of Damst
### Single DAMST Angle Analysis Phase  
As shown in Figure 3, for a single DAMST angle, the acute angle shown in Figure 3a involves a longer detour than the obtuse one shown in Figure 3b. Therefore, in the single DAMST angle analysis phase, each DAMST angle is compared with 90\({}^{\circ}\), the DAMST point whose corresponding DAMST angle is an acute angle should be cancelled from the DAMST point list. For example, if the (n\(-\)1)th DAMST angle in the DAMST angle list is less than 90\({}^{\circ}\), the (n\(-\)1)th DAMST point needs to be cancelled. For example, in Figure 4, because angles 1 and 3 are less than 90\({}^{\circ}\), P[n\(-\)1] and P[n\(-\)3] are cancelled from the DAMST point list and P[n\(-\)2] and P[n] are reserved. At the end of this phase, some DAMST points on longer detours are cancelled from the DAMST point list, accordingly, the DAMST angles in the DAMST angle list needs to be recalculated according to the reserved DAMST points based on the Definition 1. Figure 4: An example of single DAMST angle analysis. Figure 3: (**a**) Acute and (**b**) obtuse DAMST angles. B. Adjacent DAMST Angle Analysis Phase  
Subsequently, after the single DAMST angle analysis phase, a further contrastive study among adjacent DAMST angles in the DAMST angle list is executed according to the updated lists. In order to demonstrate the contrastive study, two different cases shown in Figures 5 and 6 are analyzed. In Figure 5, angles 1 and 2 are two adjacent DAMST angles in the DAMST angle list, representing the (n\(-\)1)th and (n\(-\)2)th elements in it. It can be seen that if angle 2 is greater than angle 1, the path constructed by P[n], P[n\(-\)1] and P[n\(-\)2] is smooth, no matter whether angle 2 is a clockwise angle or an anticlockwise angle. From this, if 90\({}^{\circ}\) < angle 1 < angle 2, it is assumed that the detour through P[n], P[n\(-\)1] and P[n\(-\)2] is relatively shorter and P[n\(-\)1] is supposed to be reserved in the DAMST point list. Otherwise, as shown in Figure 6, 90\({}^{\circ}\) < angle 2 < angle 1, the detour through P[n], P[n\(-\)1] and P[n\(-\)2] is assumed relatively longer than that in Figure 5, it is hereby that P[n\(-\)1] needs to be cancelled from the DAMST point list. Through comparing the adjacent DAMST angles in the DAMST angle list, more DAMST points on longer detours are cancelled. At the end of each round of the adjacent DAMST angle analysis, the DAMST angle list also needs to be updated accordingly, through recalculating DAMST angles according to the reserved nodes in the DAMST point list. Subsequently, the next round of DAMST angle analysis starts. The DAMST angle analysis is performed from one round to another until there is no more detour can be found. We now use a simple example shown in Figure 7 to illustrate how to cancel DAMST points on detours in the adjacent DAMST angle analysis phase. In Scenario A, once the mobile sink arrives at E, it appoints E as a footprint and automatically puts E into a DAMST point list. Meanwhile, the mobile sink calculates a new DAMST angle (angle 3) and puts it into DAMST angle list. After this step, the DAMST point list includes nodes B, C, D and E, and the DAMST angle list includes angle 3, angle 2 and angle 1 (all three of  
Figure 5: Trajectory with a shorter detour. Figure 6: Trajectory with a longer detour. than 90\({}^{\circ}\)). According to the adjacent DAMST angle analysis, the mobile sink needs to cancel D from the DAMST point list for angle 3 < angle 2. After canceling D, the mobile sink updates its DAMST point list (including nodes B, C and E) and DAMST angle list (including angle 4 and angle 1, both of them are greater than 90\({}^{\circ}\)). For angle 4 > angle 1 > 90\({}^{\circ}\), the DAMST angle analysis is finished at E, as shown in Scenario B. The remaining nodes in the updated DAMST point list are B, C and E, which can be used to establish a detour-aware path A-B-C-E from RA(A) to E, which is shorter than A-B-C-D-E.