Graphical analytic algorithm for designing transitional sections of transport highways

Transitional sections of transport highways are required to have a smooth curvature change along the entire length of the section. Modern design methods assume the use of both traditional transition curves and algebraic splines of high order of smoothness. In this case, a transition region consisting of several different curves is formed. In this paper, we propose a method for constructing transition curves, based on the special properties of quadratic birational transformations of a plane with coinciding fundamental points. In contrast to the known methods, applying of the method of quadratic birational transformations makes it possible to obtain a transition curve in the form of a single algebraic curve of the fourth order, instead of a composite curve consisting of sections of different curved lines. In order to determine the coefficients of the transition curve equation, the method of graphic differentiation of an implicitly set function of two variables is used. It is shown that when using modern computer graphics, the error in determining the curvature radius of the desired curve does not exceed 0.6 %. The developed graphical analytic algorithm can be used when designing transitional sections of highways and sports routes.