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

DMS-0138411 Rida T. Farouki虽然出于实际工程的考虑，曲面相交问题在数学的各个领域都有很深的根源，包括代数，几何，拓扑，分析和数值方法。 拟议的项目将带来不同的专业知识和数学家，计算机科学家和工程师的跨学科团队的新承诺，承担这个关键问题。由于成功的解决方案将需要全新的方法和范式，而不是渐进式的进步，因此将特别强调从第一原则重新审视问题，通过具有不同专业领域的参与者之间的协同对话寻求新的见解，并使用探索性软件实现作为研究工具。拟议的研究的一个中心主题是从头开始将拓扑信息纳入交叉算法的制定和验证。虽然未能保证拓扑一致性是现有算法的主要不足，这仍然是一个探索不足的领域。我们最初的研究将集中在使用同伦或“嵌入”方法来跟踪隐式定义的代数曲线的拓扑变化，因为它们从“简单”的初始配置演变成规定的相交轨迹。现代计算机辅助设计（CAD）系统的可靠性和可靠性已严重受损的持续缺乏严格的算法拓扑-曲面相交和修剪曲面的一致表示。这个基本问题的解决方案可以产生巨大的经济影响，消除目前在分析，制造和检测应用中使用CAD数据之前的费力的“几何准备和修复”任务，并解决不同CAD系统之间的数据交换不兼容性。由此带来的生产力效益，体现在加快设计周期和更多地依赖“虚拟原型”进行设计验证，对于保持技术领先地位至关重要。

DMS-0138411Rida T. FaroukiThough 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, computerscientists, 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 ofexpertise, 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 willfocus 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 thisfundamental 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.