Accuracy and Stability of Computational Representations of Swept Volume Operations

一个严格的，有效的可计算的理论分析扫描体积（一套所占的物体在空间中移动）是在这个项目中开发。 工作涉及 数学和计算科学的计算拓扑学专业。我们的目标是为扫描体操作的自动计算和表示做出根本性的贡献，并探索工程和科学中及时问题的推广和应用。这将导致新的方法和见解，推进国家的最先进的计算机辅助几何设计和相关领域。传统方法将与先进的数学技术相结合，以实现目标，其中包括以下任务：（1）开发算法，更平滑的表示扫掠体，包括有效的计算表征的准确性和稳定性，并将这些结果扩展到更一般的对象。(2)创建用于计算和表示扫描体积的布尔运算的新方法，这些方法是实时可执行的，并且包括精度和稳定性算法。(3)比较新的算法与当前的立体几何方法，并开发有效的方法来接口这些方法与现有的软件。(4)研究组织工程和虚拟造型等领域中出现的问题的应用。扫描体积在设计、制造和健康科学中具有重要的应用。一个基本的问题，关于扫描体和其他对象的表示是，如何才能确保基于计算机的表示是准确的，特别是当数据包含错误，当不同的方法被用来渲染对象？这个问题，涉及计算机生成的对象的形状，提供了该项目的动机和主要焦点。在某些情况下，有容易计算的数量表征形状，可以包括在程序中表示对象。这些量，特别是当它们应用于扫描体积时，将被深入研究，沿着获得几何对象的更平滑表示的方法。将研究开发的方法在组织工程和虚拟设计和制造中的应用，并预计在计算科学中的其他应用。该项目的研究结果将用于创建计算机辅助设计和工程应用的创新程序，并将通过出版物，讲座，网络，现有和新的本科生和研究生课程，计算拓扑学的新课程以及与科学和工程合作者的互动进行传播。

A rigorous, effectively computable theory for the analysis of swept volumes (the set occupied by objects moving in space) is developed in this project. The work involves mathematical and computing sciences in the specialty of computational topology. The goals are to make fundamental contributions to the automated computation and representation of swept volume operations, and explore generalizations and applications to timely problems in engineering and science. This will lead to new approaches and insights that advance the state-of-the-art in computer aided geometric design and related fields. Traditional methods will be integrated with advanced mathematical techniques to achieve the goals, which encompass the following tasks: (1) Develop algorithms for smoother representation of swept volumes that include effectively computable characterizations of accuracy and stability, and extend these results to more general objects. (2) Create new methods for computing and representing Boolean operations for swept volumes that are real-time executable, and include accuracy and stability algorithms. (3) Compare the new algorithms with current solid geometry methods, and develop efficient ways to interface these approaches with existing software. (4) Investigate applications to emerging problems in such areas as tissue engineering and virtual sculpting. Swept volumes have important applications in design, manufacturing and the health sciences. A basic question concerning the representation of swept volumes and other objects is, how can one insure that computer based representations are accurate, especially when the data contains errors, and when different methods are used to render the object? This question, which concerns the shape of computer-generated objects, provides the motivation for and the main focus of the project. In some cases there are readily computable quantities characterizing shape that can be included in programs for representing objects. Such quantities, especially as they apply to swept volumes, are to be studied in depth, along with methods for obtaining smoother representations of geometric objects. Applications of the approaches developed to tissue engineering and virtual design and manufacturing will be investigated, and additional applications in the computing sciences are anticipated. Findings from the project will be used to create innovative programs for computer aided design, and engineering applications, and will be disseminated via publications, lectures, the Web, existing and new undergraduate and graduate courses, new programs in computational topology, and interactions with scientific and engineering collaborators.