# Exploring Benefits of Using Blending Splines as Transition Curves
## 4 Results and Discussions
### Bezier as Local Curves  
Using Bezier curves as local curves and starting with a default ERB-function for blending, the curvature for the three blending spline transition curves together with the clothoid, and the blending spline in \([p_{3},q_{3}]\) from Figure 12, can be seen in Figure 17. As before, the dashed function is the clothoid curvature, while the solid functions are blending spline curvatures. In addition, a blending spline (dotted) with local arc curves in \([p_{3},q_{3}]\) was plotted for comparison. From the figure, it can be seen that using Bezier curves as local curves, instead of arcs, yielded blending splines with curvatures that lay closer to the clothoid, while still having a smooth transition to the end points. With this observation,  
Figure 16: Curvature plots of the clothoid with parts of the circular segments (dashed) and blending splines in \([p_{3},q_{3}]\) for balance parameters \(\lambda=0.5\) (black, default), \(\lambda=0.2\) (blue), and \(\lambda=0.8\) (red). Figure 14: Curvature plots of the clothoid with parts of the circular segments (dashed) and blending splines in \([p_{3},q_{3}]\) for tightening parameters \(\alpha,\gamma=1,1\) (black, default), \(\alpha,\gamma=4,1\) (blue), and \(\alpha,\gamma=10,1\) (red). Figure 15: Curvature plots of the clothoid with parts of the circular segments (dashed) and blending splines in \([p_{3},q_{3}]\) for slope parameters \(\beta=1.0\) (black, default), \(\beta=8.0\) (blue), and \(\beta=50.0\) (red). we concluded that the three first criteria from Section 3.3 were fulfilled, and that there is a possibility that more criteria are met. In Figure 18 the first (solid) and second (dotted) derivative of the curvature for the blending spline in \([p_{1},q_{1}]\) are plotted, together with the first derivative of the curvature for the clothoid (dashed). From the figure, it can be observed that the first derivative of the blending spline went to zero in the ends, while the second derivative ended a little above and below zero in the two end points. This indicates that criterion 4 was fulfilled, but not criterion 5, as there will be a small jump in the connection points for the second derivative function. However, the blending spline with local Bezier curves was still better than the clothoid for smoothness in the connection points to the adjacent segments. One method of increasing the smoothness, so that it also fulfills criterion 5, could be to raise the degree of the Bezier curves from \(d=4\) to \(d=5\), and fix the first four control points. The same assumption is made here, as in the discussion with arc curves, that any blending spline with a smoother curvature diagram towards the end points, than the default ERB-function blending spline, will fulfill criteria 1-4 in Section 3.3. Thus, the focus when varying the ERB parameters will be on smoothing the curvature along the blending spline. In the variation of the ERB parameters, the curves that were studied had entry points \([p_{1},q_{1}]\), because the curvature lay closest to the clothoid in Figure 17. The parameters were varied in the same manner as with the blending spline with local  
Figure 17: Curvature plots of the clothoid with parts of the circular segments (dashed) and default ERB-function blending splines, with local Bézier curves, for the three entry points \([p_{1},q_{1}]\) (blue), \([p_{2},q_{2}]\) (red), and \([p_{3},q_{3}]\) (black). In addition, the curvature of a blending spline with local arc curves, in \([p_{3},q_{3}]\), is plotted (dotted). Figure 18: Plots of the first derivative of the curvature for the clothoid (dashed) and default ERB-function blending splines, with local Bézier curves, in \([p_{1},q_{1}]\) (solid), and the second derivative of the curvature for the blending spline, scaled by \(1/10\) (dotted). arc curves, described in Table 2. In Figure 19, the tightening parameters were varied; in Figure 20, the slope parameter was varied; and in Figure 21, the balance parameter was varied.