AF: Small: Topological Graph Theory Revisited: With Applications in Computer Graphics

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Recent research in computer graphics has shown that the classical topological graph theory provides a solid mathematical foundation and powerful tool for the development of 3D modeling systems. Despite its initial success, there is still a significant gap between the theoretical research in topological graph theory and its direct applications in computer graphics. In particular, the research in classical topological graph theory has largely neglected geometric issues. Moreover, very recent research has shown exciting connections between graph embeddings on non-orientable surfaces and surface weaving, and demonstrated a need for a refined and extended study in this direction. This proposed research, guided by its applications in computer graphics, will refine and extend the classical topological graph theory in the following two directions: 1. Graph embeddings on orientable surfaces with geometric constraints: This project will re-examine the fundamental issues studied in graph embeddings on orientable surfaces that are related to topologically robust 3D modeling, by considering geometric constraints such as symmetry, planarity and conical properties. Applications of this research include development of topologically robust and highly interactive graphics modeling systems.2. Graph embeddings on non-orientable surfaces and their applications in modeling surface weaving: This project will refine and extend the study on graph embeddings on non-orientable surfaces and the corresponding graph surgery operations, study their relations to 3D modeling, and build a new paradigm of modeling surface weaving. Applications of this research include creating beautiful shapes such as woven basket and topological sculptures.