## Appendix A Geometric Perspective on Surveillance Area  
The objective of this appendix section is to expand on Formulas (1) and (2) to achieve a more comprehensive understanding. We have modified Figure 2 to Figure 12 and have provided a detailed explanation of (1) and (2).  
Figure 12 represents the initial surveillance area in the horizontal plane without taking into consideration the camera's viewing angle \(\theta_{h}\). In (1) and (2), the variable \(x\) is treated as an element of a one-dimensional array that contains values ranging from 0 to \(d_{t}\) (this decision was made to set an appropriate data type that can handle continuous and discrete values at the program implementation stage).  
As demonstrated in Figure 12, the line \((H_{x_{0}},H_{y_{0}})\) that passes through the center of the surveillance area lies along the x-axis (i.e., \(H_{y_{0}}\) is 0 for all \(H_{x_{0}}\)). When the camera's viewing angle is applied, determining \((H_{x_{0}},H_{y_{0}})\) in this state is equivalent to a problem in which the linear equation \(y=0\) is rotated for all \(x\) by \(\theta_{h}\), and the equation for \(x\) and \(y\) is expressed in an orthogonal coordinate system using trigonometric functions.  
In other words, the linear equation \(y=0\) for all \(x\) represents a straight line that is horizontal to the x-axis. If this line is rotated by theta about the origin, the equation of the rotated straight line can be obtained. Through the employment of a rotation transformation, the original value \((x,y)\) rotates by \(\theta\) and moves to a new point \((x^{\prime},y^{\prime})\). The formula for rotation transformation is as follows:  
\[x^{\prime} =x\cos(\theta)-y\sin(\theta)\] \[y^{\prime} =x\sin(\theta)+y\cos(\theta)\]  
If any point is selected on the original straight line \(y=0\), the coordinates of the point will be \((x,y)=(x,0)\). Substituting this into the rotation conversion equation:  
\[x^{\prime} =x\cos(\theta)-0\sin(\theta)=x\cos(\theta)\] \[y^{\prime} =x\sin(\theta)+0\cos(\theta)=x\sin(\theta)\]  
Thus, the equation of a straight line that rotates the original straight line \(y=0\) by \(\theta\) comprises all points \((x^{\prime},y^{\prime})\) that satisfy \(x^{\prime}\) and \(y^{\prime}\).  
Next, deriving the coordinates \((H_{x_{1}},H_{y_{1}})\) and \((H_{x_{2}},H_{y_{2}})\) that relate to the two border lines makes use of \(H_{FOV}(x)\) values and coordinates \((H_{x_{0}},H_{y_{0}})\) that relate to the center line of the surveillance area. Here, \(H_{FOV}(x)\) (represented by a purple dotted line in Figure 12) is the ratio of the width of the image sensor \(L_{sw}\) to the focal length \(L_{f}\) from the camera to the image sensor, and this is multiplied by the variable \(x\) for distance, as mentioned above in Section 3.2. That is, \(H_{FOV}(x)\) denotes the maximum width of an image that can be captured at a distance of \(x\) from the image sensor.  
Note that \(H_{FOV}(x)\) is always orthogonal to \((H_{x_{0}},H_{y_{0}})\) in an orthogonal coordinate system, as illustrated in Figure 11. Thus, it can be defined that the coordinates values \((H_{x_{1}},H_{y_{1}})\) and \((H_{x_{2}},H_{y_{2}})\) corresponding to the two boundaries of the surveillance area are rotated by \(\theta_{h}\) after shifting \((H_{x_{0}},H_{y_{0}})\) by half of \(H_{FOV}(x)\) to \((H_{x_{0}},H_{y_{0}})\). To put it another way, substituting this into the formula yields the following:  
\[x^{\prime} =x\cos(\theta)\pm\frac{1}{2}H_{FOV}(x)\sin(\theta)\] \[y^{\prime} =x\sin(\theta)\mp\frac{1}{2}H_{FOV}(x)\cos(\theta)\]