# Exploring Benefits of Using Blending Splines as Transition Curves
## 4 Results and Discussions
### Arc as Local Curves  
Curvature plots for the clothoid and three blending spline transition curves (with entry points \([p_{1},q_{1}]\), \([p_{2},q_{2}]\), and \([p_{3},q_{3}]\)), using circular arcs as local curves and a default ERB-function for blending, can be seen in Figure 12. The dashed function is the clothoid curvature, while the solid functions are blending spline curvatures (\([p_{1},q_{1}]\) in blue, \([p_{2},q_{2}]\) in red, and \([p_{3},q_{3}]\) in black). From the figure, it can be seen that changing the entry points did not have a great impact on the behaviour of the blending spline. The main difference was the length of the curves (\(L=60\) m for \([p_{1},q_{1}]\), \(L=120\) m for \([p_{2},q_{2}]\), and \(L=180\) m for \([p_{3},q_{3}]\)), which arose from the selection of entry points. Figure 11: Curvature diagram of a clothoid connected to two arc segments of different radii. Figure 12: Curvature plots of the clothoid with parts of the circular segments (dashed) and default ERB-function blending splines, with local arc curves, for the three entry points \([p_{1},q_{1}]\) (blue), \([p_{2},q_{2}]\) (red), and \([p_{3},q_{3}]\) (black). Compared to the clothoid's linear curvature, the blending splines have five inflection points in their curvature graphs, which is not optimal and should be attempted smoothed out. A positive characteristic of the blending splines is that they go smoothly towards the end points, where the clothoid has a discontinuity in connection to the constant curvature of the adjacent segments, as described in Figure 11. This leads us to believe that the blending spline, at least with a default ERB-function, satisfies the additional criteria (4 and 5 presented in Section 3.3) when compared to the clothoid. To test our hypothesis, graphs of the first and second derivatives of the curvature for the blending spline in \([p_{3},q_{3}]\) were printed. As the behaviour was fairly familiar for the three blending splines in the end points only one was checked. The result can be seen in Figure 13, where the constant first derivative of the clothoid's curvature (dashed) is plotted against the first (solid) and second (dotted) derivatives of the blending spline's curvature. From the figure, it can be observed that the first and second derivatives of the blending spline go to zero in the ends. These conditions are exactly what is needed to fulfill criteria 4 and 5. Hence, the blending spline is better than the clothoid when it comes to smoothness in the connection points to the adjacent segments. From the previous results, an assumption can be made that any blending spline with a smoother curvature diagram towards the end points, than the default ERB-funtion blending spline, will meet all the criteria in Section 3.3. Thus, the focus of the discussion when varying the ERB parameters will be on smoothing the curvature along the blending spline. In the adjustment of the ERB parameters, the curves that are studied have entry points \([p_{3},q_{3}]\). By varying the tightening parameters, \(\alpha\) and \(\gamma\), as described in Table 2, the resulting curvature diagrams, of the blending splines, become as shown in Figure 14. From the plots, it can be observed that by increasing \(\alpha\), the smoothness in the end points increased and the bumps became more prominent, with higher amplitudes of the extreme values. By varying the slope parameter, \(\beta\), as described in Table 2, the resulting curvature diagrams, of the blending splines, became as plotted in Figure 15. From the plots, it can be observed that by increasing \(\beta\), the bumps were shifted toward the middle of the graph and the smoothness increased in the end points. By varying the balance parameter, \(\lambda\), as described in Table 2, the resulting curvature diagrams, of the blending splines, became as plotted in Figure 16. From the plots, it can be observed that by moving the balance of the ERB-function left or right, the smoothness increased in one end and decreased in the other.