Mathematical description of the transitional part of the railway line plan

The most complex part of a railway track plan is its transition curve. The transition curve is the section of the track plan that connects the circular arc portion of the railway layout to the straight section of the track. It serves to provide a smooth transition from the straight section to the circular arc of a given radius. The radius of the circular section is determined based on the angle between the connecting straight segments of the railway. A mathematical model of the railway, depending on the angle and radius of curvature, is provided in [1]. In this article, we determine the optimal method for defining the length of the transition curve. This length is computed under the assumption that the transition section is a clothoid.