The geometry description plays a central role in many engineering applications and directly influences the quality of the computer simulation results. The geometry of a space curve can be completely defined in terms of two parameters: the horizontal and vertical curvatures, or equivalently, the curve curvature and torsion. In this paper, distinction is made between the track angle and space-curve bank angle, referred to in this paper as the Frenet bank angle. In railroad vehicle systems, the track bank angle measures the track super-elevation required to define a balance speed and achieve a safe vehicle operation. The formulation of the track space-curve differential equations in terms of Euler angles, however, shows the dependence of the Frenet bank angle on two independent parameters, often used as inputs in the definition of the track geometry. This paper develops the general differential equations that govern the track geometry using the Euler angle sequence adopted in practice. It is shown by an example that a curve can be twisted and vertically elevated but not super-elevated while maintaining a constant vertical-development angle. The continuity conditions at the track segment transitions are also examined. As discussed in the paper, imposing curvature continuity does not ensure continuity of the tangent vectors at the curve/spiral intersection. Several curve geometries that include planar and helix curves are used to explain some of the fundamental issues addressed in this study.

几何描述在许多工程应用中起着核心作用，并直接影响计算机模拟结果的质量。空间曲线的几何形状可以完全由两个参数定义：水平曲率和垂直曲率，或者等效地，曲线曲率和扭转。在本文中，轨道角和空间曲线坡度角之间进行了区分，在本文中称为Frenet 坡度角。在铁路车辆系统中，轨道倾斜角的措施跟踪超-海拔需要定义一个平衡速度并实现车辆安全运行。然而，根据欧拉角的轨道空间曲线微分方程的公式显示了Frenet 坡度角对两个独立参数的依赖性，通常用作轨道几何定义中的输入。本文使用实践中采用的欧拉角序列开发了控制轨道几何形状的一般微分方程。举例说明，在保持恒定的垂直-展开角的情况下，曲线可以扭曲和垂直升高但不能超升高. 还检查了轨道段过渡处的连续性条件。正如论文中所讨论的，施加曲率连续性并不能确保曲线/螺旋交点处切向量的连续性。包括平面曲线和螺旋曲线在内的几种曲线几何形状用于解释本研究中解决的一些基本问题。