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Special Issue: Computer Graphics and Multimedia

• Regular Paper •

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

Hao Guan1,2,3, Yong-Sheng Rao1,*, Member, CCF, Jing-Zhong Zhang1, Sheng Cao4, Member, CCF, and Xiao-Lin Qin2,3, Member, CCF

1. 1 Institute of Computing Science and Technology, Guangzhou University, Guangzhou 510006, China;
2 Chengdu Institute of Computer Applications, Chinese Academy of Sciences, Chengdu 610041, China;
3 University of Chinese Academy of Sciences, Beijing 100049, China;
4 School of Computer Science and Engineering, University of Electronic Science and Technology of China Chengdu 61173, China
• Received:2019-10-10 Revised:2021-01-15 Online:2021-07-05 Published:2021-07-30
• Contact: Yong-Sheng Rao E-mail:rysheng@gzhu.edu.cn
• About author:Hao Guan is a postdoctor researcher in Institute of Computing Science and Technology, Guangzhou University, Guangzhou. He received his Ph.D. degree in computer science from University of Chinese Academy of Sciences, Beijing, in 2020. His research interests include computer graphics and educational software.
• Supported by:
This work is supported by the Sichuan Science and Technology Program of China under Grant Nos. 2018GZDZX0041 and 2020YFG0011, the National Natural Science Foundation of China under Grant No. 11701118, the Guangzhou Academician and Expert Workstation under Grant No. 20200115-9, and Key Disciplines of Guizhou Province of China—Computer Science and Technology under Grant No. ZDXK [2018]007.

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.

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