# Global Time-Varying Path Planning Method Based on Tunable Bezier Curves
## 3 Transient Path under the Single-Obstacle Environment
### Feature Points of Curves  
The feature points of the obstacle are extracted in Step 4 and the feature points of the curve are extracted in Step 5. In this step, the obstacle is magnified with the \(O_{0}\) point as the center point. The enlarged boundary is used to obtain the points \(P_{i}^{0}\) by which the Bezier curve is drawn and the line between these points is a Bezier polygon. In this paper, the magnification scale \(k\) is set to 2 and the points on the obstacle surface are regarded as set \(E\) and the points on the magnified boundary are regarded as set \(K\) and then the two sets satisfy the following relationship.  
\[K=k\cdot E-O_{0} \tag{6}\]It can be specifically shown in the image that the line segments in Figure 1 satisfy the relationship shown in Equation (7):  
\[\left\{\begin{array}{l}\overline{P_{2}^{0}O_{0}}=k\cdot\overline{x_{\text{ left}}O_{0}}\\ \overline{P_{3}^{0}O_{0}}=k\cdot\overline{E_{1}O_{0}}\\ \overline{P_{4}^{0}O_{0}}=k\cdot\overline{E_{2}O_{0}}\\ \overline{P_{5}^{0}O_{0}}=k\cdot\overline{x_{\text{right}}O_{0}}\end{array}\right. \tag{7}\]  
The line segments in Figure 2 satisfy the relationship shown in Equation (8):  
\[\left\{\begin{array}{l}\overline{P_{2}^{0}O_{0}}=k\cdot\overline{x_{\text{ left}}O_{0}}=k\cdot\overline{E_{1}O_{0}}\\ \overline{P_{3}^{0}O_{0}}=k\cdot\overline{E_{2}O_{0}}\\ \overline{P_{4}^{0}O_{0}}=k\cdot\overline{E_{3}O_{0}}\\ \overline{P_{5}^{0}O_{0}}=k\cdot\overline{x_{\text{right}}O_{0}}=k\cdot \overline{E_{4}O_{0}}\end{array}\right. \tag{8}\]  
The line segments in Figure 3 satisfy the relationship shown in Equation (9):  
\[\left\{\begin{array}{l}\overline{\frac{P_{3}^{0}O_{0}}{P_{4}^{0}O_{0}}=k \cdot\overline{E_{1}O_{0}}}\\ \angle P_{2}^{0}P_{1}^{0}=d_{\text{start}}\\ \angle P_{6}^{0}P_{5}^{0}=d_{\text{end}}\end{array}\right. \tag{9}\]  
The points in Figure 4 satisfy the relationship shown in Equation (10):  
\[\left\{\begin{array}{l}P_{2}^{0}=(\min(\mathbf{K},x),0)\\ P_{3}^{0}=(\min(\mathbf{K},x),\max(\mathbf{K},y))\\ P_{4}^{0}=(\max(\mathbf{K},x),\max(\mathbf{K},y))\\ P_{5}^{0}=(\max(\mathbf{K},x),0)\end{array}\right. \tag{10}\]  
where \(\min(\mathbf{K},x)\) is the minimum value of the set \(\mathbf{K}\) in the \(x\)-axis direction.  
In summary, in addition to finding set \(\mathbf{K}\) of the enlarged boundary points according to set \(\mathbf{E}\) and the magnification scale \(k\), the following steps are included in Step 5:  
1. Set the start point \(x_{\text{start}}\) as the first point in feature points of the curve and the end point \(x_{\text{end}}\) as the last points in feature points of the curve;
2. Make a straight line through the \(x_{\text{start}}\) point in the direction \(d_{\text{start}}\) (start direction) and find the solution in set \(\mathbf{K}\) (set \(\mathbf{K}\) is the set of points on the boundary after the enlarged surface of the obstacle), which is the second point in the characteristic point of the curve;
3. Make a straight line through the \(x_{\text{end}}\) point in the direction \(d_{\text{end}}\) (end direction) and find the solution in set \(\mathbf{K}\), which is the penultimate point in the feature point of the curve;
4. Amplify the \(E_{i}\) of the part in which the trajectory planning needs to be developed and the enlarged point is regarded as a feature point of the curve (such as Figures 1-3), or the maximum value point of the coordinate is regarded as a feature point of the curve (such as Figure 4).