This study consists on finding the optimal transition curves to ensure a smooth transition of the projection of the car and railway axis on the horizontal plane from the straight parts to the circular parts or the front part of the two circular parts. The optimization is performed using the selection of expressions defining the transition curve and the dependence of its length on the angle of rotation. During the optimization, all boundary conditions are strictly followed so that the radius of curvature at the transition points is the same for both curves. The expression of the transition curve is derived from the solution of the differential equation of the curvature as a special case of the inverse problem. To design the required sections of limited-length transition curves of the road, the expression derived from the differential equation of curvature was improved and a new equation for the transition curve was introduced. The equations representing the transition curve reflect a smooth transition from a straight section to an arc of a circle. A sequence of construction, a mathematical apparatus and a new method for constructing a road transition line using computer technology have been developed. A table has been compiled to determine the length of a transition curve section depending on the Cartesian coordinate system of the road. Electronic experimental tests are used with the application of graphical programs to provide a graphical representation of the curvature along the entire transition curve. It is proved by the graphical representation of the curvature obtained from the electronic experiment that there is no jump at the transition points and along the entire curve, i.e., the velocity and centrifugal force change monotonically along the projected transition line.
Imanov et al. (Thu,) studied this question.