# Exploring Benefits of Using Blending Splines as Transition Curves
## 1 Introduction  
Transition curves are an important development in the railway industry. They were introduced to create easements between straight and curved railway sections, and between curved sections of different radii. Properly constructed transition curves increased safety and comfort while travelling, and reduced wear on the rail [1, 2]. Various curves have been recommended as transition curves over the years, among them, the clothoid [3, 4], which has become the most used to this day. The search for new types of transition curves for railways is still relevant and ongoing, due to new knowledge related to curve properties [5], where the linear curvature of the clothoid is not smooth enough in the end regions to be optimal. Both in relation to passenger comfort, for speeds above 120 km/h [6], and in relation to wear on the tracks with heavy haul traffic. The recent research on transition curves can be divided into linear- and non-linear curvature curves. The most common topic of the two is non-linear curvature curves, since the problem of increasing curvature smoothness has received attention. Some examples are a remodelling of the cubic parabola [7], a new design of the Bloss curve [8], sinusoid transition curves [9], and the Wiener Bogen curves [10]. Under the topic of linear curvature curves, two types of curves have been analysed: log-aesthetic curves [6], which are identified to give equal properties as clothoids, and Symmetrically Projected Transition Curves [11], which are simpler and more accurate curves compared to the clothoidand cubic parabolas, and are meant as a better option in cases where the cubic parabola is preferred over the clothoid. In addition, experiments have been performed on hybrid solutions, e.g., curves with linear middle parts and non-linear ends. Two examples are parametric transition curves [12] and smoothed transition curves [13; 14]. All of the previously mentioned curves are curves that replace a single element in the railway. However, there is also research where multiple elements are replaced. General transition curves [15; 16] and universal transition curves [17; 18] are curves that replace the segments: first transition curves-circular arc-second transition curve, or first transition curves-second transition curve. Multiple element transition curves are not investigated in this paper, but are interesting for future work. A common denominator with the research on transition curves is that there is rarely a connection between research and industry, as can be observed in [19]. A challenge here is replacing segments in an already existing railway. Recent research does not take into account the layout of the new transition curves and the effects they will have on the existing railway, in regard to the horizontal alignment of the rail. A blending spline curve [20] is a spline construction where local functions are blended together by \(C^{k}\)-smooth blending functions. Blending splines were introduced [21] as an additional tool in computer-aided geometric design (CAGD) with emphasis on user interactivity via the editing capabilities. Due to the flexibility in the blending process, when connected to possible local functions and blending functions, the authors suggest that the spline may be suitable as a transition curve. The scope of this study was to find a new railway transition curve, with better properties than the clothoid in regard to the smoothness between segments, that can replace the clothoid in an already existing railway without making changes to the original alignment. The need to be able to increase smoothness without making extensive changes to the alignment is relevant for existing railways where large modifications may be invasive or impossible. One example is Ofotbanen, a Norwegian railway line, essential in the transportation of ore from Kiruna in Sweden, and goods between the northern and southern parts of Norway [22]. Ofotbanen is known for its challenging railway geometry, which is vulnerable to wear.