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

AF：小：拓扑图论重温：计算机图形学中的应用

• 批准号: 0917288
• 负责人: Jianer Chen
• 金额: $ 38.67万
• 依托单位: Texas A&M Engineering Experiment Station
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2009
• 资助国家: 美国
• 起止时间: 2009-09-01 至 2013-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0917288&HistoricalAwards=false
• 关键词: AF; Small; Topological; Graph; Theory

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.

计算机图形学的最新研究表明，经典拓扑图理论为三维造型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，但拓扑图论的理论研究与其在计算机图形学中的直接应用之间仍有很大的差距。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，不可定向曲面上的图嵌入和曲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更精细和更广泛的研究。这项研究以其在计算机图形学中的应用为指导，将从以下两个方向提炼和扩展经典拓扑图理论：1.几何约束可定向曲面上的图形嵌入：本项目将通过考虑对称性、平面性和圆锥性等几何约束，重新研究可定向曲面上图形嵌入的基本问题，这些基本问题与拓扑稳健的三维建模有关。本研究的应用包括开发具有拓扑性的健壮和高度交互的图形建模系统。不可定向曲面上的图形嵌入及其在曲面编织建模中的应用：本项目将完善和扩展不可定向曲面上的图形嵌入和相应的图形运算的研究，研究它们与三维建模的关系，并建立建模曲面编织的新范式。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。