Computational Topology and its Applications in Geometric Design

Background: In recent geometric design, many effective toolkits for geometric modeling and optimization have been proposed and applied in practical cases, while effective and efficient designing of shapes that have desirable topological properties remains to be a challenge. The development of computational topology, especially persistent homology, permits convenient usage of topological invariants in shape analysis, geometric modeling, and shape optimization. Persistence diagram, the useful topological summary of persistent homology, provides a stable representation of multiscale homology invariants in the presence of noise in original data. Recent works show the wide use of persistent homology tools in geometric design. Objective: In this paper, we review the geometric design based on computational topological tools in three aspects: the extraction of topological features and representations, topology-aware shape modeling, and topology-based shape optimization. Methods: By tracking the development of each aspect and comparing the methods using classical topological invariants, motivations, and key approaches of important related works based on persistent homology are clarified. Results: We review geometric design through topological extraction, topological design, and shape optimization based on topology preservation. Related works show the successful applications of computational topology tools of geometric design. Conclusion: Solutions for the proposed core problems will affect the geometric design and its applications. In the future, the development of computational topology may boost computer-aided topological design.