## 4 Conclusions  
This method could either output geometric models or numeric data. After the solution process, the resulting axes could be directly used for modeling. Alternatively, in a collaborative workflow including many sessions, to make data exchange convenient, only grid spacing and rotation angle are required. Both of them are numbers, and therefore easy to store in a general format such as.txt,.xlsx, etc. The grid spacing is used to create a grid system in BIM software such as Autodesk Revit and Bentley Openbuildings Designer. The rotation angle is used to rotate the point cloud to fit the grid system; thereafter, the point cloud is used as a reference during the modeling process.  
Although this method aims to automate the solution process, regarding the diversity of architectures there are still manual operations. For example, removing some columns based on the standard column grids is a common practice, such as Kuiwen Pavilion. Not all axis intersections have columns; therefore, the corresponding column has to be removed from the algorithm.  
Furthermore, this method could be applied to test whether these columns are designed to be inclined or not. First, the axes of column bases are generated. The centroid of rebuild axes on column bases could be recorded as C0, and the centroid of rebuild axes on column heads could be recorded as C1. The vector from C0 to C1 on the XY plane would indicate the movement between the column heads grid system and the column bases grid system. If the column heads grid system is moved to make the X and Y coordinates of C1 and C0 consistent, the result would indicate whether these columns are designed to be inclined or not (Figures 19 and 20).  
If they are designed to be inclined, the outer eaves columns are inclined inward in the front and back eaves, with 10% of the column height; 8% of the column height in the two sides eaves, and the corner columns are inclined in both directions at the same time.  
After the manual check, the result shows that the inclination of the columns almost conforms to a designed inclination, as the average distance is 50 mm, the column height is 5027 mm, and the ratio is almost 10%. However, this result still needs further investigation to confirm, as all of the columns have inclinations, rather than just eaves columns. This practice has not been found in historical archives or architectural books. The cause of this special inclination needs further research to verify whether this construction method of column inclination is common in other Imperial-style architectures.  
Conceptualization, X.W., C.W. and C.B.; Funding acquisition, C.W.; Investigation, X.W. and C.W.; Methodology, X.W., C.W. and C.B.; Project administration, C.W.; Resources, C.W.; Software, X.W.; Validation, X.W.; Visualization, X.W.; Writing--original draft preparation, X.W.; Writing--review and editing, X.W. and C.W. All authors have read and agreed to the published version of the manuscript.  
This research was funded by Ministry of Science and Technology of the People's Republic of China (project No. 2020 YFC1522400, No. 2020 YFC1522405).  
Not applicable.  
The source code of the node-visible algorithm is not openly available due to the consideration of data safety.  
Figure 19: The inclination and the average distance.  
Figure 20: The column height.  
**Acknowledgments:** This paper was partially supported by the following institutions: The Qufu Administrative Bureau of Historical Relics, Qufu; The Qufu San Kong Heritage Architecture Engineering Management Office, Qufu; The Palace Museum, Beijing; Jiayuguan Academy of Silk Road (the Great Wall) Culture, Jiayuguan; Cultural Heritage Administration of Jizhou District, Tianjin.  
**Conflicts of Interest:** The authors declare no conflict of interest.