CARGO: Advanced Topological Methods for Robust Surface Intersection Algorithms and Trimmed Surface Representations

University Of California-Davis, Davis CA

Investigators

Abstract

DMS-0138411 Rida T. Farouki Though motivated by practical engineering considerations, surface intersection problems have deep roots in diverse areas of mathematics, including algebra, geometry, topology, analysis, and numerical methods. The proposed project will bring the diverse expertise and renewed commitment of an inter-disciplinary team of mathematicians, computer scientists, and engineers to bear on this critical problem. Since successful solutions will require radically new approaches and paradigms, rather than incremental advances, special emphasis will be placed on re-visiting the problem from first principles, seeking new insights through synergistic dialog among participants with diverse areas of expertise, and use of exploratory software implementations as research tools. A central theme of the proposed research is the ab initio incorporation of topological information into the formulation and verification of intersection algorithms. Although failure to guarantee topological consistency is the main deficiency of existing algorithms, this remains a poorly explored domain. Our initial line of enquiry will focus on using the homotopy or "embedding" method to track topological changes in implicitly-defined algebraic curves, as they evolve from "simple" initial configurations into a prescribed intersection locus. The accuracy, efficiency, and reliability of modern computer-aided design (CAD) systems has been severely compromised by the persistent lack of rigorous algorithms for topologically-consistent representations of surface intersections and trimmed surfaces. The solution to this fundamental problem can have tremendous economic impact, in terms of eliminating the laborious "geometry preparation and repair" task that currently precedes the use of CAD data in analysis, manufacturing, and inspection applications, and in resolving data-exchange incompatibilities among different CAD systems. The ensuing productivity benefits, manifested in accelerated design cycles and greater reliance on "virtual prototyping" for design verification, are crucial to the maintenance of technological leadership.