# Exploring Benefits of Using Blending Splines as Transition Curves
## 5 Concluding Remarks  
In this paper, a family of splines (blending splines) were applied and analysed as transition curves for an already existing railway. Three segments from the Ofotbanen railway were extracted as two circular arcs connected by a clothoid. The goal was to make the least amount of intervention to the existing railway, by keeping the placement of the arcs unchanged and only allowing the transition curve entry points to move. Under this constraint, the blending splines were compared to the clothoid with regard to the smoothness of the end points and the shape of the curvature graph. The main findings in the paper can be summarized as follows:  
* the blending spline transition curves had a higher degree of curvature smoothness (\(\geq G^{1}\)) in the connection points compared to the clothoid, which was \(G^{0}\);
* the blending spline yielded the possibility of replacing transition curves in an existing railway without moving the adjacent segments;
* the choice of blending function and local curves had a large impact on the curvature of the curve, and on the degree of smoothness in the connection points. The results showed that the blending splines, with appropriately selected local curves, were smoother than the clothoid for the ends. However, the curvature plots of the blending splines had fluctuations along the lengths, which may not be optimal in a transition case if the fluctuations are large. An observation was made that the choice of ERB parameters and local curves had a large impact on these fluctuations. The choice of local curve type and shape had an even larger impact. By using Bezier curves as local curves, with a default ERB-function, the fluctuations were relatively small, as can be observed in Figure 17. The curvature function was close to a monotonically increasing function, which seems to be the norm for transition curves. However, there exists at least one transition curve that is not, the Wiener Bogen [10]. The Wiener Bogen uses a non-monotonically increasing curvature together with a monotonically increasing superelevation ramp, which provides better transition properties than the clothoid. Whether the blending spline curves can be used in the same manner as the Wiener Bogen remains to be tested, and may yield interesting results if the blending splines are smoother in the end points when compared with the clothoid. A different approach, which is very relevant for future work, is to test other local curves that may dampen the fluctuations more (or even remove them). From the results on local Bezier curves, it will be of interest to raise the degree of the Bezier so that more than one control point in the ends can be manipulated. There are strong indications that these extra degrees of freedom can be utilized to smooth the curvature function more than for fourth degree Bezier curves where only one control point is manipulated. In this work, the control point was moved manually in order to test the hypothesis. However, there are indications that it would be possible to develop an algorithmic solution for the placement of  
Figure 21: Curvature plots of the clothoid (dashed) and blending splines in \([p_{1},q_{1}]\) for balance parameters \(\lambda=0.5\) (black, default), \(\lambda=0.2\) (blue), and \(\lambda=0.8\) (red). the control points in the local Bezier curves, under the constraints of a given optimal curvature. This, together with inspecting ERB-functions where several parameters are modified, with respect to the characteristics of the resulting curvature, is out of scope in the current paper and is considered for future work. From a technical point of view, the local curves of the proposed transition curves can be represented in already existing software, as circular arcs and Bezier curves are fundamental elements in CAD. Utilizing blending spline transition curves with established workflows and software tools can then be performed in several ways.