Research on the Movement Patterns and Optimal Turning Paths of Bench Dragon Troupes Based on Geometric Models

This paper focuses on the mathematical modeling and turning optimization of the intangible cultural heritage art "Bench Dragon," with an emphasis on the kinematic analysis of the dragon's head and body, collision detection within the formation, determination of the maximum travel time, and optimization of the minimum turning radius. By integrating methods from analytical geometry, numerical solutions of differential equations, optimization models, and algorithms, a multidisciplinary analytical framework was constructed. Through dynamic simulation and iterative algorithms, the movement trajectory of the Bench Dragon's body was precisely reconstructed. A novel rectangular collision detection model was proposed to mathematically define the concept of "bench collision," and the maximum safe travel time for the formation was determined to be 414 seconds. Using the bisection method, the possible range for the minimum turning radius was hypothesized, and based on optimization algorithms, the interval length was gradually reduced to meet the precision requirement of 0.01, ultimately solving for the minimum turning radius of 0.423 meters. An extendable bisection method program was developed to achieve rapid solutions. The research results provide quantitative analysis tools for the digital preservation of traditional arts, and the algorithmic framework offers methodological reference value for the study of group motion systems.