ITR: Accurate and Robust Operations on Curved Geometry

Texas A&M Engineering Experiment Station, College Station TX

Investigators

Abstract

Curved geometry plays an important role in many applications, including manufacturing, robotics, and graphics. In many cases, highly accurate and robust operations and representations are desired, but the perceived implementation difficulties and inefficiencies (e.g. from exact computation) prevent application. Compared to the linear and combinatorial problems traditionally associated with computational geometry, problems involving curved solids are far more complicated. In practical terms, problems with accuracy, efficiency, and robustness are all significantly worse when dealing with curved geometry, and significant work remains in addressing these three areas. With further development, efficient implementations, and demonstrated applications, however, accurate and robust operations will find greater acceptance, giving a potential benefit to several application areas. This proposal outlines a plan of research that addresses accuracy, efficiency, and robustness of operations on curved geometry. Overall, it is a topic that is both challenging and promising in terms of potential impact and future research opportunities. The proposed research addresses fundamental areas needed for further development of precise operations with curved solids. Theoretical issues are addressed, but a primary focus of the proposed research is on implementation. Most of the areas of proposed research are geared toward finding highly accurate or exact implementations that still achieve reasonable efficiency. Software will be released to support dissemination of the results. Specific areas of research include: - Filtered Sequences of Computations - Incorporating Root-Finding into Expression Trees - Filtered Geometric Representations - Resultant Computations - Evaluating Combinations of Efficiency Improvements - Handling Degeneracies - Computing with Transcendental Functions - Exact Geometric Representations Application will be a major goal for all of the research areas. Application will serve both to drive new research ideas, and to allow testing and evaluation of research results. The primary application that will be used is solid modeling, from a traditional CAD perspective. The need for accurate and efficient representations in solid modeling is well recognized, making this research particularly relevant. In addition, solid modeling applications present a wide range of problems of varying complexity that allow thorough evaluation of the new methods. Besides solid modeling, we will also investigate application to geometric modeling of other objects, such as neurons, plants, and other naturally occurring objects. Application to neuron modeling will be in conjunction with an ongoing project on mapping brain microstructure, an exciting new interdisciplinary problem area with enormous potential that Texas A&M is in a unique position to address. The proposed research will be integrated into educational activities. A new course in practical computer algebra will be developed that supports the research areas. Undergraduate student research will be encouraged through direct involvement in projects, coursework, and interaction through service activities.