## 2 Evaluation Index of Segmentation Accuracy
### Profile Fitting Degree  
The segmentation accuracy of the compressed trajectory is evaluated by the trajectory profile fitting degree, and the definition is given as follows. **Definition 4**. : _The profile fitting degree is the fitting degree between the compressed trajectory and the original trajectory after the trajectory is linearly approximated to a set of straight lines, which is recorded as \(OF\). It is affected by the projection angle factor and the projection distance factor. The projection angle factor is a function of the projection length of the original trajectory and the length of the original trajectory. It describes the influence of the angle change of the approximate line segment on the similarity, which is recorded as \(OF_{ang}\). The projection distance factor is a function of the projection length of the original trajectory and the distance from the original trajectory to the approximate line segment, which describes the influence of the distance difference between the approximate line segments on the similarity and is recorded as \(OF_{dis}\). Figure 1 shows an example of the linear approximation of the original trajectory segment \(TRSub_{1}=\left\{P_{1},P_{2},\ldots,P_{8}\right\}\) to \(TRSub_{1}^{\prime}=\left\{P_{1}^{\prime},P_{8}^{\prime}\right\}\)._The projection of the original trajectory sub segment \(P_{1}P_{2}\) is \(P_{1}^{\prime}P_{2}^{\prime}\), the projection of the sub segment \(P_{2}P_{3}\) is \(P_{2}^{\prime}P_{3}^{\prime}\), etc., all together constitute the approximate trajectory segment \(TRSub_{1}^{\prime}\). For the original sub segment \(P_{i}P_{i+1}\), the projection angle factor and the projection distance factor are defined as  
\[OF_{ang}(P_{i}P_{i+1})=d(P_{i}^{\prime}P_{i+1}^{\prime})/d(P_{i}P_{i+1}) \tag{1}\]  
\[OF_{dis}(P_{i}P_{i+1})=\left\{ \begin{array}{l}1-e^{-d(P_{i}^{\prime}P_{i+1}^ {\prime})/d_{Min}(P_{i}P_{i+1})},d_{Min}(P_{i},P_{i+1})>0\\ 1,d_{Min}(P_{i},P_{i+1})=0\end{array} \right. \tag{2}\]  
where \(d(P_{i}P_{i+1})\) is the distance of the original trajectory segment \(P_{i}P_{i+1}\), \(d(P_{i}^{\prime}P_{i+1}^{\prime})\) is the distance of the projection \(P_{i}^{\prime}P_{i+1}^{\prime}\), and \(d(P_{i}^{\prime}P_{i+1}^{\prime})\leq d(P_{i}P_{i+1})\). \(len_{S}\) is the length of the original trajectory sub segment and \(0\leq i\leq len_{S}-1\). We define \(d_{Min}(P_{i}P_{i+1})=\min[d(P_{i}P_{i}^{\prime}),d(P_{i+1}P_{i+1}^{\prime})]\), and the solution of \(d(P_{i}P_{i}^{\prime})\) is based on Helen formula, as shown in Figure 1. The area of \(\Delta P_{1}^{\prime}P_{i}P_{8}^{\prime}\) is  
\[S_{\Delta}=\sqrt{p(p-a)(p-b)(p-c)}=\frac{1}{2}\cdot h\cdot c \tag{3}\]  
where \(p\) is the half perimeter, \(a,b,c\) is the side length \(d(P_{1}P_{i}),d(P_{8}P_{i}),d(P_{1}P_{8})\), and \(h\) is \(d(P_{i}P_{i}^{\prime})\). Obviously, \(OF_{ang}(P_{i}P_{i+1})\) describes the influence of the angle \(\theta(0\leq\theta\leq\pi/2)\) between \(P_{i}P_{i+1}\) and \(P_{i}^{\prime}P_{i+1}^{\prime}\) on the degree of trajectory approximation. The smaller the angle is, the closer the distance of \(P_{i}P_{i+1}\) and \(P_{i}^{\prime}P_{i+1}^{\prime}\) is, and tends to 1, while the larger the angle is, \(OF_{ang}(P_{i}P_{i+1})\) tends to 0. In \(OF_{dis}(P_{i}P_{i+1})\) calculation, \(d(P_{i}P_{i}^{\prime})\) and \(d(P_{i+1}P_{i+1}^{\prime})\) are the distance from the two endpoints of the original sub segment to the projection, and \(d_{Min}(P_{i},P_{i+1})\) takes the smaller value of both, because the difference between them reflects the change of angle, which has been reflected in the calculation of \(OF_{ang}(P_{i}P_{i+1})\). It can be seen that \(OF_{dis}(P_{i}P_{i+1})\) describes the influence of distance on trajectory approximation. The smaller the distance is, the closer \(P_{i}P_{i+1}\) and \(P_{i}^{\prime}P_{i+1}^{\prime}\) are, and \(OF_{dis}(P_{i}P_{i+1})\) tends to 1. The larger the distance is, the farther away \(P_{i}P_{i+1}\) and \(P_{i}^{\prime}P_{i+1}^{\prime}\) are, and \(OF_{dis}(P_{i}P_{i+1})\) tends to 0.