## 3 Modeling Methodology for Railway Track Structure Representation
### Representation of Discontinuous Railway Track Structures  
The creation of shapes and attributes of discontinuous objects is an important step. The methodology depicted in Figure 3 was used to represent discontinuous structures. First, discontinuous objects that were to be arranged along the recreated railway alignment were modeled using BATs. The alignment was regenerated by applying BATs to the alignment information, which was generated using AMTs; the locations of the discontinuous objects were then calculated. Figure 3: Algorithm for shape representation of discontinuous structures. Figure 2: Process of railway track modeling using software linkage. AMT: Alignment-centered modeling tools; BAT: BIM authoring tools; IFC: STEP Tools; IFC: industry foundation class. \(L(t,i)\), which is the rule for correctly placing discontinuous objects, is defined in Equation (1). \[L(t,i):=p(t,i)\wedge r(t,i), \tag{1}\]  
where \(p(t,i)\) is the conditional function for placing a discontinuous object \(i\) at point \(t\) along the alignment, and \(p(t,i)=c_{i}\longmapsto t_{i}\) holds; \(c_{i}\) denotes a specific point that represents object \(i\); and \(\longmapsto\) indicates a mapping process. \(r(t,i)\) represents the rotation condition of object \(i\) at point \(t\). The rotation condition is required for placing discontinuous objects perpendicular to the direction of the alignment and has an equivalent operational relationship with \(v_{i}\). The object rotation vector \(v_{i}\) can be expressed using Equation (2). \[v_{i}=\left[\theta_{1}^{i};\theta_{2}^{i};\theta_{3}^{i}\right], \tag{2}\]  
where \(\theta_{1}\), \(\theta_{2}\), and \(\theta_{3}\) represent rotation angles (radians) based on the \(x\)-, \(y\)-, and \(z\)-axes, respectively. In this study, the direction of alignment was set to the \(y\)-direction, and the direction perpendicular to the geometric surfaces was set as the \(z\)-direction. Therefore, \(\theta_{1}\) represents the pitch, \(\theta_{2}\) the roll, and \(\theta_{3}\) the yaw. \(\theta_{3}\) follows the change in alignment in the \(xz\)-plane. Thus, \(\theta_{3}\) can be calculated using Equation (3). \[\theta_{3}=\left\{ \begin{array}{cc}\cos^{-1}\left(\frac{d_{t}^{r}\cdot u}{|d _{t}^{r}|}\right)&d_{t}(1)\geq 0\\ -\cos^{-1}\left(\frac{d_{t}^{r}\cdot u}{|d_{t}^{r}|}\right)&d_{t}(1)<0\end{array}  \right., \tag{3}\]  
where \(d_{t}^{r}=[d_{t}(1);d_{t}(2);0]\) and \(u=[0;1;0]\) holds. \(d_{t}\) is the direction vector at point \(t\) on the alignment. \(u\) is the local reference axis of object \(i\) and is set to the \(y\)-axis in this study. \(\theta_{1}\) is the rotation angle for the slope of the alignment and is calculated using Equation (4). \[\theta_{1}= \begin{cases}\cos^{-1}\left(\frac{d_{t}d_{t}^{r}}{|d_{t}^{r}|} \right)&d_{t}(3)\geq 0\\ -\cos^{-1}\left(\frac{d_{t}d_{t}^{r}}{|d_{t}^{r}|}\right)&d_{t}(3)<0\end{cases} , \tag{4}\]  
where \(d_{t}^{r}=[d_{t}(1);d_{t}(2);0]\) holds. \(\theta_{2}\) has the same meaning as the cant in railway alignment because it represents the rotation value around the direction of the alignment. In other words, this value is calculated using a separate equation depending on the radius of curvature of the alignment. In this study, the value created by the AMT was used for the BAT. The AMT creates a cant value at points where the alignment equation changes, and this value is calculated via a linear function. \(\theta_{2}\) can be calculated using the cant value, as shown in Equation (5). \[\theta_{2}^{k(i)}=\sin^{-1}\left(\frac{\Delta_{2}^{k(i)}}{m_{i}}\right), \tag{5}\]  
where \(\Delta_{2}^{k(i)}\) is the cant value of object \(i\) and can be calculated by Equation (6). \[\Delta_{2}^{k(i)}=\frac{\delta^{k(i)+1}-\delta^{k(i)}}{t_{k(i)+1}-t_{k(i)}} \times\left(c_{i}\sim t_{k(i)}\right)+\delta^{k(i)}, \tag{6}\]  
where \(k(i)\) represents the cant critical point--that is, the point at which the cant change begins in the alignment to which object \(i\) belongs; in other words, the value of \(k\) changes at the point where the alignment equation changes.