COLLABORATIVE RESEARCH PROPOSAL: Hybrid Modeling for Geometric Design, Estimation, and Analysis

这是犹他州大学和北卡罗来纳州大学在教堂山的合作研究项目。该项目通过构建一组新的曲面表示来推动几何建模的发展，这些曲面表示足够通用，可以解决几何曲面设计、估计和分析中的各种问题。这项工作的基础是断言，没有一个单一的表示不足以有效和准确地解决现代应用中出现的无数计算和分析问题。因此，策略是开发一个系统地结合参数曲面表示和隐式曲面表示的框架。本项目将研究两种特殊表示的互补性：非均匀B-样条线(NURBS)和水平集。它将开发方法，提供：表示的相互跟踪，分析发生的拓扑和奇点，以及在表示之间交流重要的拓扑事件的机制。本项目还将研究这一框架在计算机辅助设计以及地形建模和分析方面的应用。当使用曲面对3D对象进行建模时，人们面临着各种不同的曲面表示。每种表现形式都有各自的缺点和优点。许多建模应用程序，如计算机辅助设计和地形分析，都需要修改形状以匹配某些输入数据或满足用户或设计者的需求。因此，表面表示的最重要的方面之一是它如何为用户或计算机算法提供修改形状的便利，即将一个形状变形为另一个略有不同的形状。一些表示为用户提供了大量的控制，但被限制为有限的特定形状集。其他表示法可以表示各种形状，但精度有限。修改形状的一个重要问题是表示如何处理曲面折叠回来并接触自身的情况。这被称为奇点，它表明曲面可能正在从一类形状过渡到另一类形状。不同的曲面表示以不同的方式处理这种奇点。这项工作试图通过结合两种不同的形状表示和扩展相关的奇点理论来改进形状修改和奇点处理的技术。这种探索应该会产生用于设计形状的新工具，以及用于构建和分析测量表面形状的新工具，例如从地形生成的形状。犹他州(031075)和北卡罗来纳大学教堂山分校(0310546)的合作为这个项目提供了奇点理论方面的专门技能和独特的教育机会，让计算机科学专业的学生更多地了解这一重要的数学学科。

ABSTRACTThis is a collaborative research project between the Universities of Utah and North Carolina at Chapel Hill. The project is advancing the state of the art in geometric modeling by constructing a set of new surface representations that are sufficiently general to solve a wide range of problems in geometric surface design, estimation, and analysis. This work is founded on the assertion that no single representation is adequate to solve efficiently and accurately the myriad computational and analytical problems arising in modern applications. The strategy, therefore, is to develop a framework that systematically combines parametric and implicit surface representations. This project will examine the complementary nature of two particular representations: non-uniform B-splines (NURBs) and level sets. It will develop methods which provide: mutual tracking of the representations, analysis of the topology and singularities which occur, and mechanisms for communicating important topological events between representations. This project will also examine applications of this framework to computer aided design and terrain modeling and analysis. When modeling 3D objects with surfaces, one is faced with a wide range of different surface representations. Each representation has corresponding drawbacks and advantages. Many modeling applications, such as computer-aided design and terrain analysis, require shapes to be modified in order to match some input data or to suit the needs of a user or designer. Therefore, one of the most important aspects of a surface representation is how it affords a user or a computer algorithm the facility to modify a shape, i.e. to deform one shape into another slightly different shape. Some representations provide a user with a great deal of control, but are constrained to a limited specific set of shapes. Other representations can represent a wide range of shapes, but only with a limited accuracy. An important concern in modifying a shape is how the representation handles cases where the surface folds back and touches itself. This is called a singularity and it indicates that the surface might be transitioning from one class of shapes to another. Different surface representations handle such singularities in different ways. This work will try to improve the technology for modifying shapes and handling singularities by combining two different shape representations and extending relevant singularity theory. This exploration should yield new tools for designing shapes and new tools for building and analyzing the shapes of measured surfaces, such as those generated from terrains. The collaboration between Utah (031075) and the University of North Carolina Chapel Hill (0310546) provides this project with specialized skills in singularity theory and unique educational opportunities for computer science students to learn more about this important subject in mathematics.