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

GOALI/IUCP：处理以非均匀有理 B 样条 (NURBS) 形式表示的几何图形的算法方法

• 批准号: 9526119
• 负责人: Leslie Piegl
• 金额: $ 24.1万
• 依托单位: University of South Florida
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1996
• 资助国家: 美国
• 起止时间: 1996-11-01 至 2000-10-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9526119&HistoricalAwards=false
• 关键词: GOALI; IUCP; Algorithmic; Approach; Processing

*** 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. ***

*9526119 Piegl这笔资金用于开发新的几何框架和算法，用于表示、设计和处理非均匀有理B-Splines(NURBS)几何图形。开发的算法将为自由形状几何所覆盖的工程零件的稳健和准确的设计和处理提供工具。重点是利用有理形式的额外自由度来开发新的技术，用于插补和逼近、形状修改、光顺以及各种基本算法，如符号算子和升阶和降阶。为了利用B-Splier方法的优点，降低B-Spline的复杂度，提出了一种B-Spline-to-B-Spline方法。算法将在现有NURBS框架的环境中实现，该框架具有用于所有支持计算工作的工具。工业伙伴将在其建模系统的框架内进行涉及所开发算法的效率和有用性的实验。还将根据大多数CAD/CAM系统中的规范建立和评估一个实验系统。完成后，这项研究的结果将导致NURBS理论的改进和高级算法的开发。预计在与所提出的算法相关的样条理论和相关工具，如符号算子、有理样条的插补和逼近、数据简化、曲线和曲面光顺以及结点和权重理论方面将获得新的理论结果。新的高级几何算法也将出现，要么作为新理论结果的直接实现，要么作为问题的直接算法解决方案，例如最优节点布置。还将引入分解-贝塞尔-重组范式，以更有效地解决一大组问题，如升阶和降阶、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Rational形式的力量，以在数据存储、灵活性和速度方面获得更高效的算法。高效的算法将缩短交货期，增强现有系统的健壮性、可靠性和效率，并最终提高发动机和涡轮叶片等产品的质量。NURBS的Bezier范例将有助于理解许多复杂的算法，因此将有助于跨各种工程学科标准化NURBS。***