Creation of road and railway routes in general and also design of vertical alignment in particular constitute an significant engineering problem. The vertical alignment of roads, and especially of railways, is of great importance for passenger comfort under high-speed driving conditions. This requires alignment smoothness to a greater extent. In practice, the dominant tendency is to consider vertical arcs as parabolic curves (quadratic or cubic). Other solutions have been also proposed and analyzed in the literature, thus providing an alternative approach to traditional methods. This paper describes a design method of vertical arcs using the so-called general transition curves. These curves describe the entire curvilinear transition between two straight lines using a single equation, and the curvature at the contact points with the straight lines is zero, which is of great practical significance in the light of vehicle motion dynamics. As a result, relevant research shows that general transition curves, compared to other curves, provide a more favorable distribution of lateral changes of acceleration. The paper deals not only with appropriate design procedures of a general nature, but also specific procedures of fitting vertical arcs to preset points. The methods for designing vertical arcs constitute a useful extension of available methods, and offer a valuable tool for engineers and designers of road and railway routes.

中文翻译：

---

使用一般过渡曲线设计公路和铁路垂直线形的曲线部分

一般而言，公路和铁路路线的建设以及特别是垂直线形的设计构成了一个重要的工程问题。道路，特别是铁路的垂直线形对于高速行驶条件下乘客的舒适度非常重要。这就需要更大程度的对准平滑度。在实践中，主要趋势是将垂直弧视为抛物线（二次或三次）。文献中还提出并分析了其他解决方案，从而提供了传统方法的替代方法。本文描述了一种使用所谓的一般缓和曲线的垂直弧的设计方法。这些曲线用一个方程描述了两条直线之间的整个曲线过渡，并且与直线接触点的曲率为零，这对于车辆运动动力学具有重要的实际意义。因此，相关研究表明，与其他曲线相比，一般过渡曲线提供了更有利的加速度横向变化分布。本文不仅涉及一般性质的适当设计程序，而且还涉及将垂直弧拟合到预设点的具体程序。设计垂直弧的方法构成了现有方法的有用扩展，并为公路和铁路路线的工程师和设计师提供了宝贵的工具。