Topology Understanding and Topology Control for 3D Models by Computational Topology: A Survey

The principal objective of Computer-Aided Geometric Design (CAGD) is the design and processing of 3-dimensional (3D) geometric models, of which topological structure and geometric shape are two key components. Historically, however, CAGD has prioritized geometric design and processing, lacking a systematic methodology for understanding and controlling the topological structure of 3D models. As the application scope of modeling technology expands, challenges related to topology understanding and topology control in model design and processing have become increasingly prominent. These challenges are particularly evident in domains such as microstructure and molecular structure design. Consequently, there is an urgent need to develop systematic and effective theories and technologies for both topology understanding and control. This survey explores how computational topology, or Topological Data Analysis (TDA), with persistent homology as its core analytical tool , addresses the critical challenges of topology understanding and topology control for 3D geometric models. We review the theoretical foundations, computational methods, and practical applications across various domains. The survey demonstrates that a systematic methodology based on persistent homology has been developed for the topological design and processing of 3D geometric models, signaling the emergence of a new discipline: Computer-Aided Topological Design (CATD).