GOALI/IUCP: Algorithmic Approach to Processing Geometry Represented in Non-Uniform Rational B-Splines (NURBS) Form

* 9526119皮格尔 该GOALI赠款为开发新的几何框架和算法提供资金，用于非均匀有理B样条（NURBS）几何的表示，设计和处理。所开发的算法将提供强大的和准确的设计和加工的自由形式的几何覆盖的工程零件的工具。重点是利用额外的自由度的合理形式，开发新的技术，插值和近似，形状修改，光顺，和各种基本算法，如符号运算符，和度升降和减少。提出了一种B样条到Bezier到B样条的方法，以充分利用Bezier方法的优点并降低B样条的复杂性。这些算法将在现有的NURBS框架中实现，该框架具有用于所有支持计算工作的工具。涉及开发的算法的效率和实用性的实验将由工业合作伙伴在其建模系统的框架内进行。还将根据大多数CAD/CAM系统中的规范构建和评估实验系统。 完成后，这项研究的结果将导致改进的NURBS理论和先进的算法开发。预计新的理论成果将在样条理论和相关的工具，如符号运算符，插值和近似与有理样条，数据减少，曲线和曲面光顺，并在理论的结和权重有关的建议的算法。新的高级几何算法也将出现，要么作为新的理论结果的直接实现，要么作为问题的直接算法解决方案，如最佳结的位置。一个分解-贝塞尔-重组范式也将被引入，以更有效地解决大量的问题，如度升降，重新参数化，和各种近似。这项工作的主要目标是更深入地了解理性形式的力量，以获得在数据存储，灵活性和速度方面更有效的算法。高效的算法将缩短交货时间，增强现有系统的鲁棒性、可靠性和效率，并最终提高发动机和涡轮机叶片等产品的质量。NURBS的Bezier范例将有助于理解许多复杂的算法，因此将有助于在各种工程学科中标准化NURBS。 ***

*** 9526119 Piegl This GOALI grant provides funding for the development of a new geometrical framework and algorithms for the representation, design and processing of Non-Uniform Rational B-splineS (NURBS) geometry. The developed algorithms will provide tools for the robust and accurate design and processing of engineering parts covered by free-form geometry. The emphasis is on utilizing the extra degrees of freedom of the rational form to develop new techniques for interpolation and approximation, shape modification, fairing, and various fundamental algorithms such as symbolic operators, and degree elevation and reduction. A B-spline-to-Bezier-to-B-spline approach is also proposed to take advantage of Bezier methods and to reduce the complexity of B-splines. The algorithms will be implemented in the context of an existing NURBS framework that has tools for all supporting computational work. Experiments involving the efficiency and usefulness of the developed algorithms will be performed by the industrial partner within the framework of its modeling system. An experimental system will also be built and evaluated according to specifications found in most CAD/CAM systems. Upon completion, the results of this research will lead to improvements in both NURBS theory and advanced algorithm development. It is anticipated that new theoretical results will be obtained in spline theory and associated tools such as symbolic operators, interpolation and approximation with rational splines, data reduction, curve and surface fairing, and in the theory of knots and weights as related to the proposed algorithms. New advanced geometric algorithms will also emerge either as direct implementations of new theoretical results or as direct algorithmic solutions of problems such as optimal knot placement. A decomposition-Bezier-recomposition paradigm will also be introduced to more efficiently solve a large set of problems such as degree elevation and reduction, reparametrization, and various approximations . The primary goal of this work is to gain a deeper insight into the power of the rational form to obtain more efficient algorithms in terms of data storage, flexibility and speed. Efficient algorithms will cut lead time, enhance the robustness, reliability and efficiency of existing systems, and will ultimately improve the quality of products such as engines and turbine blades. The Bezier paradigm of NURBS will help in understanding many complicated algorithms and hence will contribute to standardizing NURBS across various engineering disciplines. ***