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

这项拨款为开发相对简单和创新的数学程序和计算机算法提供资金，用于建模、对准、修改和测量具有自由形状或雕刻形状的大型制造物体。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发展中发挥重要作用。该算法将用于描述自由形式曲线的基元；确定任意构型物体的姿态；对现有隐式多项式模型进行局部和全局修正；并在平行平面的二维曲线之间插入雕刻的三维曲面。最先进的三坐标测量机将用于许多拟议的研究，以测试，完善和验证程序。如果成功，这项研究将为具有复杂轮廓的物体（如涡轮叶片，齿轮，船体和汽车车身）的建模和测量提供更多用户友好的方法；简化了计算机辅助各种制成品的几何设计程序；使用三坐标测量机在更短时间内更精确地测量这些物体的新技术；以及逆向工程算法的简化，以生产没有已知技术图纸或规格的对象。这将降低成本，提高未来制造产品的质量。此外，潜在的数学分解将导致对更高次代数曲线的更好理解，这是一个基本的数学主题，自17世纪以来，一些世界上最重要的数学家已经广泛研究了它。代数曲线广泛应用于许多不同的实际技术领域，增加对其潜在特性的了解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.