NUMBER: 0635008 INSTITUTION: Ohio State University Research Foundation PRINCIPAL INVESTIGATORS: Dey, Tamal K. & Wenger, R. S. Collaborative Proposal NUMBER: 0635366 INSTITUTION: University of Illinois PRINCIPAL INVESTIGATORS: Ramos, Edgar
PROJECT TITLE: Collaborative Research: Non-smoothness in Meshing and Reconstruction
The problems of meshing and reconstruction are pervasive in science and engineering where geometric domains need to be digitally represented, analyzed, inspected, or prototyped. Although the input forms to these two problems differ, both have a similar goal in producing a triangular representation of a geometry. Considerable theoretical advances have been recently made in designing and implementing provable algorithms for both. However, these algorithms assume some form of smoothness of the input domain. As a result, current solutions are not broad enough to handle many of the geometric domains which arise in scientific studies and engineering applications. Designing machine parts in automotive industry, creating virtual environments with buildings, simulating cracks and shocks in scientific studies are a few examples with non-smoothness as a basic impediment. This project studies the difficulty of non-smoothness in meshing and reconstruction by designing sound algorithms and by developing robust software based on these algorithms.
Meshing produces a triangulation of an explicitly specified geometric domain whereas reconstruction does the same with a point sample. Most of the provable reconstruction algorithms exploit the differential structure of a smooth surface whereas meshing algorithms, though allowing polyhedral domains, restrict the input angles not to be small. The result of these restrictions is that non-smooth domains such as piecewise smooth surfaces and non-manifolds cannot be handled with full generality. The goal of this project is to broaden the class of input geometry for which models can be computed with assurance of accuracy. The research in this project uses concepts from various mathematical disciplines such as differential geometry, differential topology, and non-smooth analysis and also tools from areas of theoretical computer science such as computational geometry, computational topology, and numerical optimization. Graduate students supported by the project develop skills in theoretical computer science, most notably in computational geometry and topology and also in writing robust, efficient and user-friendly software.
