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

这是犹他州和北卡罗来纳州在查佩尔山的大学之间的合作研究项目。该项目通过构建一组新的表面表示来推进几何建模的最新技术，这些表面表示足够通用，可以解决几何表面设计，估计和分析中的各种问题。 这项工作是建立在断言，没有一个单一的表示是足以有效地解决和准确地在现代应用中出现的无数的计算和分析问题。 因此，该战略是开发一个框架，系统地结合参数和隐式表面表示。 这个项目将研究两种特殊表示的互补性：非均匀B样条（Non-uniform B-splines，简称B-splines）和水平集。 它将开发的方法，提供：相互跟踪的表示，分析的拓扑结构和奇点发生，并表示之间的通信机制的重要拓扑事件。 本项目还将研究该框架在计算机辅助设计和地形建模与分析中的应用。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。 每种代表都有相应的缺点和优点。 许多建模应用程序，如计算机辅助设计和地形分析，需要修改形状，以匹配某些输入数据或满足用户或设计师的需求。 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工具，即将一个形状变形为另一个稍微不同的形状。 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。 其他表示可以表示各种形状，但精度有限。修改形状时的一个重要问题是表示如何处理曲面折叠并接触自身的情况。 这被称为奇点，它表明表面可能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异性。 本文将通过结合两种不同的形状表示和扩展相关的奇异性理论，来改进形状修改和处理奇异性的技术。 这种探索应该产生新的工具来设计形状和新的工具来构建和分析测量表面的形状，例如从地形生成的形状。犹他州（031075）和北卡罗来纳州查佩尔山（0310546）的大学之间的合作提供了这个项目与奇点理论的专业技能和独特的教育机会，为计算机科学的学生学习更多关于这个重要的数学科目。

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.