# Concealed Conduit Routing in Building Slabs
## 3 Results
### Test Case 1  
The appearance of the first building is shown in Figure 7a. Its dimensions, slabs, and openings are depicted in Table 1. It has four walls, one ceiling, and one floor. The thickness of these slabs is 30 cm. In the experiment, the slabs are decomposed into \(1\) cm \(\times\) 1 cm \(\times\) 1 cm voxels. Thus, a slab contains 10 layers of voxel planes. We use this model to simulate the piping process in a simple factory. Four concealed pipes are to be routed in the building. Their basic parameters and applications are listed in Table 2. They are used for transferring electricity, gas, water, and air. Their diameters are 3, 6, 8, and 3 cm, respectively. The cross-sections of these pipes occupy at least 9 voxels and at most 64 voxels. In order to meet safety regulations, a pipe must be separated from other pipes by a certain distance. This distance is called the safe gap, which is enforced by using \(\epsilon_{1}\) in Equation (1). This safe gap is maintained in only the plane of the slab that the pipe penetrated. Hence, it is a 2D distance regulation. If we select a large \(\epsilon_{2}\) when creating the feasible spaces, the feasible spaces are enlarged and allow the pipes to deviate from existing obstacles and hazards further, and more space will be engaged in the routing calculation. The safe gaps of the pipes are shown in the fifth column of Table 2. \begin{table}
\begin{tabular}{c c c c} \hline
**Dimension** & **Slabs** & **Opening** \\ \hline Width = 8 m, & 4 walls (8 m \(\times\) 4 m), & 1 door (1.8 m \(\times\) 3 m), \\ Length = 8 m, & 1 floor (8 m \(\times\) 8 m), & 7 windows (2 m \(\times\) 2.6 m) \\ Height = 4 m & 1 ceiling (8 m \(\times\) 8 m) & \\ \hline \end{tabular}
\end{table} 
Table 1: Geometric information of the first test model. \begin{table}
\begin{tabular}{c c c c c c} \hline
**Pipe** & **\#(Terminals)** & **Diameter** & **Usage** & **Safe Gap** & **Color** \\ \hline
1 & 4 & 3 cm & Electricity & 20 cm & Green \\ \hline
2 & 3 & 6 cm & Natural gas & 10 cm & Orange \\ \hline
3 & 6 & 8 cm & Water & 5 cm & Blue \\ \hline
4 & 2 & 3 cm & Air & 2 cm & Red \\ \hline \end{tabular}
\end{table} 
Table 2: Pipe parameters in test case 1. Figure 7: Routing concealed pipes in a single-room building, **(a)** the building, **(b-e)** routing pipe 1–4, **(f)** the resultant pipe systems. The progression of the routing process is shown in Figure 7b-e. In these images, the slabs are rendered as transparent grey polygons, and the pipes are shaded as solid and colorful tubes. Pipe 1 is arranged first. Its appearance is shaded green and is shown in Figure 7b. This pipe has four terminals located at the front and back walls and the ceiling. These terminals will be used for installing light bulbs and electrical switches. We use white balls to represent the light bulbs. The next pipe to be routed is pipe 2. Its path is shown in Figure 7c. We render it using orange. It has three terminals, but the resultant path contains only one branch. This is because its initial skeleton passes the third terminal, and thus no extra branch is needed to connect the third terminal. The third pipe is the water pipe. It has six terminals and its diameter is the largest. Its path is rendered in blue and displayed in Figure 7d. Its path contains many branches such that the six terminals can be successfully connected. Its branches are mainly created on the floor because all terminals except one are located there. The last pipe to be routed is the air-conditioning pipe. It has only two terminals. Its path is shown in Figure 7e. As the image shows, its source and destination are at the back and front walls. Its path has to be routed inside the ceiling to connect these two terminals since the workspace in the floor has been occupied by the third pipe. In order to display the 3D positions of these pipes in the building and to examine the quality of the pipe paths, a virtual reality image is produced to reveal the final scenario of the working environment.