Method for Processing Graph Degeneracy in Dynamic Geometry Based on Domain Design

A dynamic geometry system, as an important application in the field of geometric constraint solving, is widely used in elementary mathematics education; moreover, the dynamic geometry system is also a fundamental environment for automated theorem proving in geometry. In a geometric constraint solving process, a situation involving a critical point is often encountered, and geometric element degeneracy may occur at this point. Usually, the degeneracy situation must be substantively focused on during the learning and exploration process. However, many degeneracy situations cannot be completely presented even by the well-known dynamic geometry software. In this paper, the mechanisms causing the degeneracy of a geometric element are analyzed, and relevant definitions and formalized descriptions for the problem are provided according to the relevant modern Euclidean geometry theories. To solve the problem, the data structure is optimized, and a domain model design for the geometric element and the constraint relationships thereof in the dynamic geometry system are formed; furthermore, an update algorithm for the element is proposed based on the novel domain model. In addition, instances show that the proposed domain model and the update algorithm can effectively cope with the geometric element degeneracy situations in the geometric constraint solving process, thereby achieving unification of the dynamic geometry drawing and the geometric intuition of the user.