New Algebraic Methods for Simplifying the Design and Measurement of Sculptured Shapes

这笔赠款提供资金，用于开发相对简单和创新的数学程序和计算机算法，用于建模、调整、修改和测量具有自由形状或雕塑形状的大类人造物体。一种将二维代数曲线分解为简单直线和二次曲线乘积和的新算法将在这些新程序的开发中发挥重要作用。该算法将被用来描述自由曲线的基元；确定任意形状物体的姿态；局部和全局地修改现有的隐式多项式模型；以及在平行平面上的两维曲线之间插入雕塑的三维曲面。许多拟议的调查将使用最先进的坐标测量机来测试、改进和验证程序。如果研究成功，这项研究将提供更方便用户的方法来建模和测量具有复杂轮廓的对象，如涡轮叶片、齿轮、船体和车身；简化各种制造对象的计算机辅助几何设计程序；使用坐标测量机更准确、更短时间地测量此类对象的新技术；以及简化逆向工程算法，以生产出没有已知技术图纸或规范的对象。这将降低成本，提高未来制造产品的质量。此外，基本的数学分解将导致对高次代数曲线的更好理解，高次代数曲线是一个基本的数学问题，自17世纪以来，一些世界上最顶尖的数学家对此进行了广泛的研究。代数曲线在许多不同的实践和技术领域中被广泛使用，增加与其基本特征相关的洞察力将促进它们在这些领域的使用。

This grant provides funding for the development of relatively simple and innovative mathematical procedures and computer algorithms for modeling, aligning, modifying and measuring large classes of manufactured objects that have free-form or sculptured shapes. A new algorithm for decomposing two-dimensional algebraic curves as sums of products of simple lines and conics will play a major role in the development of these new procedures. This algorithm will be used to describe free-form curves in terms of their primitives; to determine the pose of objects in arbitrary configurations; to modify existing implicit polynomial models both locally and globally; and to interpolate sculptured, three dimensional surfaces between two dimensional curves in parallel planes. A state-of-the-art coordinate measuring machine will be used in many of the proposed investigations to test, refine and verify the proceduresIf successful, this research will provide more user-friendly methods for modeling and measuring objects with complex contours, such as turbine blades, gears, ship hulls and automobile bodies; streamlined procedures for the computer aided geometric design of a large variety of manufactured objects; new techniques for measuring such objects more accurately and in less time using coordinate measuring machines; and the simplification of reverse engineering algorithms to produce objects for which there are no known technical drawings or specifications. This will reduce the cost and improve the quality of future manufactured products. Furthermore, the underlying mathematical decomposition will lead to a far better understanding of higher degree algebraic curves, a fundamental mathematical subject that has been studied extensively since the 17th century by some of the world's most foremost mathematicians. Algebraic curves are used widely in many different practical, technical areas, and adding insight relative to their underlying features will facilitate their use in these areas.