Ancient pagodas are usually parts of hot tourist spots in many oriental countries due to their unique historical backgrounds. They are usually polygonal structures comprised by multiple floors, which are separated by eaves. In this paper, we propose a new method to investigate both the rotational and reflectional symmetry of such polygonal pagodas through developing novel geometric models to fit to the 3D point clouds obtained from photogrammetric reconstruction. The geometric model consists of multiple polygonal pyramid/prism models but has a common central axis. The method was verified by four datasets collected by an unmanned aerial vehicle (UAV) and a hand-held digital camera. The results indicate that the models fit accurately to the pagodas’ point clouds. The symmetry was realized by rotating and reflecting the pagodas’ point clouds after a complete leveling of the point cloud was achieved using the estimated central axes. The results show that there are RMSEs of 5.04 cm and 5.20 cm deviated from the perfect (theoretical) rotational and reflectional symmetries, respectively. This concludes that the examined pagodas are highly symmetric, both rotationally and reflectionally. The concept presented in the paper not only work for polygonal pagodas, but it can also be readily transformed and implemented for other applications for other pagoda-like objects such as transmission towers.
On the ineffectiveness of constant rotation in the primitive equations and their symmetry analysis
