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

近年来计算机图形学领域的研究表明，经典拓扑图理论为三维建模系统的开发提供了坚实的数学基础和有力的工具。尽管取得了初步的成功，但拓扑图理论研究与拓扑图在计算机图形学中的直接应用之间仍然存在着很大的差距。特别是经典拓扑图理论的研究在很大程度上忽略了几何问题。此外，最近的研究表明，在不可定向表面上的图形嵌入与表面编织之间存在令人兴奋的联系，并表明需要在这一方向上进行细化和扩展的研究。本研究将以其在计算机图形学中的应用为指导，从以下两个方面对经典拓扑图理论进行完善和扩展：具有几何约束的可定向表面上的图形嵌入：该项目将通过考虑对称、平面性和圆锥性质等几何约束，重新审视与拓扑鲁棒3D建模相关的可定向表面上的图形嵌入所研究的基本问题。本研究的应用包括拓扑鲁棒性和高度交互性图形建模系统的开发。非定向曲面上的图嵌入及其在曲面编织建模中的应用：本项目将对非定向曲面上的图嵌入及其相应的图手术操作的研究进行细化和拓展，研究其与三维建模的关系，构建曲面编织建模的新范式。这项研究的应用包括创造美丽的形状，如编织篮子和拓扑雕塑。

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.