Accuracy and Stability of Computational Representations of Swept Volume Operations

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.