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

rida T. farouki虽然受到实际工程考虑的启发，但表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。该计划将汇集数学家、计算机科学家和工程师组成的跨学科团队的各种专业知识和新的承诺，来解决这一关键问题。由于成功的解决方案将需要全新的方法和范例，而不是渐进式的进步，因此将特别强调从最初的原则重新审视问题，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。提出的研究的中心主题是从头开始将拓扑信息纳入交集算法的制定和验证。虽然不能保证拓扑一致性是现有算法的主要缺陷，但这仍然是一个很少探索的领域。我们最初的研究将集中在使用同伦或“嵌入”方法来跟踪隐定义代数曲线的拓扑变化，因为它们从“简单”的初始配置演变为规定的相交轨迹。现代计算机辅助设计（CAD）系统的准确性、效率和可靠性由于缺乏严格的算法来表示曲面相交和裁剪表面的拓扑一致性而受到严重损害。这个基本问题的解决方案可以产生巨大的经济影响，就消除目前在分析，制造和检查应用中使用CAD数据之前的费力的“几何准备和修复”任务而言，并解决不同CAD系统之间的数据交换不兼容性。随之而来的生产力效益，体现在加速的设计周期和对设计验证的“虚拟原型”的更大依赖上，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.