# A Semi-Random Elliptical Movement Model for Relay Nodes in Flying Ad Hoc Networks
## 4 Node Distribution in SREM  
### Numerical Analysis  
The SREM model is a mobility framework designed to effectively induce node density aligned along the communication path in FANET environments. Each node moves in  
Figure 3: Visualization of the SREM trajectories over different simulation steps. straight lines between two points located on the boundary of an ellipse. Given an ellipse with semi-major axis \(a>0\) and semi-minor axis \(b>0\), the position of a node on the boundary can be parameterized by an angular value \(\theta\in[0,2\pi)\). The \(n\)-th waypoint of UAV \(i\) is thus defined as \(P_{i}^{(n)}=\left(a\cos\theta_{i}^{(n)},\ b\sin\theta_{i}^{(n)}\right)\). At each time interval, the UAV selects the next waypoint \(P_{i}^{(n+1)}\) and moves at constant speed along a straight line from \(P_{i}^{(n)}\) to \(P_{i}^{(n+1)}\). To theoretically approximate node density along the elliptical path in SREM, we consider the infinitesimal arc length:  
\[ds(\theta)=\sqrt{a^{2}\sin^{2}\theta+b^{2}\cos^{2}\theta}\,d\theta \tag{2}\]  
Assuming uniform angular motion, UAVs tend to cluster in regions where the arc length is shorter. Thus, the node density is inversely proportional to the arc length, leading to the following approximation: \(\rho(\theta)\propto\frac{1}{ds(\theta)}\). This implies higher node concentration near the endpoints of the major axis (\(\theta=0,\pi\)) and lower density near the minor axis (\(\theta=\frac{\pi}{2},\frac{3\pi}{2}\)), supporting SREM's design objective of aligning mobility patterns with the relay path. Although this provides a rough estimate of node distribution, more accurate modeling would require incorporating detailed mathematical formulations and additional parameters. To obtain an accurate probability estimate, the probability density function \(f_{XY}(x,y)\), which represents the likelihood that a node is located at a specific position \((x,y)\), is defined as the cumulative contribution of all possible linear trajectories that pass through that point. This formulation leads to the following double integral expression:  
\[f_{XY}(x,y)\propto\int_{0}^{2\pi}\int_{0}^{2\pi}\mathrm{Ind}\bigg{[}(x,y)\in \overline{P_{i}^{(n)}P_{i}^{(n+1)}}\bigg{]}\cdot\frac{1}{\bigg{\|}_{i}^{(n)}P_ {i}^{(n+1)}\bigg{\|}}\cdot\frac{1}{L^{2}}\,d\theta_{i}^{(n+1)}\,d\theta_{i}^{( n)} \tag{3}\]  
* \(\mathrm{Ind}\bigg{[}(x,y)\in\overline{P_{i}^{(n)}P_{i}^{(n+1)}}\bigg{]}\): An indicator function that returns \(1\) if the point \((x,y)\) lies on the line segment \(\overline{P_{i}^{(n)}P_{i}^{(n+1)}}\), and \(0\) otherwise. * \(\frac{1}{\bigg{\|}_{i}^{(n)}P_{i}^{(n+1)}\bigg{\|}}\): A weighting factor inversely proportional to the segment length, reflecting that UAVs spend more time on shorter paths. * \(\frac{1}{L^{2}}\): A normalization constant assuming that departure and arrival angles are independently and uniformly sampled along the elliptical perimeter \(L\). However, due to the inclusion of the indicator function and the double integral over all possible line segments, this formulation presents significant analytical challenges. Deriving a closed-form expression for the spatial distribution at a given point becomes intractable. A similar structure is observed in the SRCM model, where a UAV selects departure and destination points randomly along a circular trajectory and moves at a constant speed. For the circular case, an approximate closed-form probability density function in polar coordinates can be derived as follows:  
\[f_{XY}(x,y)=\mu\bigg{\{}2\pi^{2}(r-x^{2}-y^{2})^{2}+4\pi\sqrt{r-(x^{2}+y^{2}) }-\frac{5}{2}\arctan\Big{(}\frac{y}{\chi}\Big{)}-\Big{[}\arctan\Big{(}\frac{y}{ \chi}\Big{)}\Big{]}^{2}\bigg{\}} \tag{4}\]  
However, as illustrated in Figure 4, the actual visualized distributions exhibit numerical instability, including asymmetry and distortion near the center or boundary--deviating from the ideal circular symmetry.