This work demonstrates how the railway plan's curved part can be modeled mathematically, highlighting the use of differential equations and curvature concepts.
The long-term practice of drawing up a railway plan is based on a curve formed by the projection of the center of the rolling stock onto a horizontal plane. At the same time, this curve is considered flat. The railway plan itself consists of straight parts and a curved part connecting these straight parts. The existence of a curved part in the railway plan is natural. The work substantiates that the railway plan in the curved part can be interpreted as a spatial curve and a natural trihedron can be constructed. With the help of this, the Frenier formula is determined. This formula, from the course on differential curve geometry, is proposed to be accepted as a mathematical model of the railway plan. Frenier's formula is a system of differential equations. Moreover, this system has a solution for a given curvature and torsion. Thus, it is possible to find the equation of the curve of the part of the railway plan.
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Artikbaev et al. (2025) studied this question.
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