Approximate Algebraic Methods in Computer Aided Geometric Design

计算机辅助几何设计中的近似代数方法

• 批准号: 8813688
• 负责人: Thomas Sederberg
• 金额: $ 18.48万
• 依托单位: Brigham Young University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1988
• 资助国家: 美国
• 起止时间: 1988-12-01 至 1992-11-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=8813688&HistoricalAwards=false
• 关键词: Approximate; Algebraic; Methods; Computer; Aided

This research is to investigate fundamental algorithms for computer aided geometric design based on algebraic methods operating in a finite precision computing environment. This proposal is motivated by the basic intersection problems in computer aided geometric design: curve-curve intersection, curve-surface intersection, and surface-surface intersection. The investigators plan to explore efficient and robust methods for computing the points of intersection between two nonplanar rational curves, the points of intersection between a rational curve and a rational surface, and the curve of intersection between two rational surface patches. Several other basic and supporting problems will also be addressed. These problems include: finding a tight upper bound on the number of intersection points between two non-planar rational curves of known degrees, determining if a rational surface patch is improperly parametrized, deciding how base points affect the degree of a parametric patch in floating point arithmetic, and developing exact global and approximate local implicitization algorithms for rational surface patches. The research will also investigate numerically stable alternatives to Euclid's Greatest Common Division algorithm for floating point arithmetic. This will also involve a combination of theoretical, empirical, and practical work. The investigators plan first to develop the mathematical theory, then to test this theory on curves and surfaces used in industrial applications, and finally to code the algorithms into practical software.

这项研究是调查的基本算法， 计算机辅助几何设计 在有限精度计算环境中运行。 这 建议的动机是基本的交叉问题， 计算机辅助几何设计：曲线-曲线相交， 曲线-曲面相交和曲面-曲面相交。 研究人员计划探索有效和强大的方法， 计算两个非平面有理函数的交点 曲线，有理曲线和 有理曲面和两个有理曲面的交线 表面补丁。 其他几个基本的和支持性的问题将 也要解决。 这些问题包括：找一个紧鞋面 两个非平面相交点个数的界 已知次数的有理曲线，确定有理曲面 面片参数化不正确，决定基点如何影响 浮点运算中参数面片的阶，以及 开发精确全局和近似局部隐式化 有理曲面片的算法 该研究还将 欧几里得最大值数值稳定性研究 浮点运算的常用除法算法。 这也将涉及理论上的， 经验和实际工作。 调查人员计划首先 发展数学理论，然后在曲线上测试这个理论 以及工业应用中使用的表面，最后对 将算法转化为实用软件。