Computational Methods for Pythagorean-hodograph Curves in CAD/CAM Applications

Pythagorean-hodograph（PH）曲线提供了许多重要的计算优势，超过了“普通”Bezier/B样条表示，通常用于CAD/CAM应用。其中的属性，区分PH曲线从后者是：在一个多项式弧长函数的精确整流;其偏移量的精确合理表示;一个封闭的形式的表达，适合于优化的目的，其弹性弯曲能量;和倾向于产生更公平的轨迹-表现出更平滑的曲线分布-在离散数据的插值。此外，现有的CAD系统可以很容易地利用这些属性，因为PH曲线是完全兼容的贝塞尔/B样条表示。 本项目研究有关PH曲线的构建和操作的突出理论和算法问题。 他们的解决方案将允许PH曲线的全部潜力在快速原型和数控铣削等应用中得以实现。这些问题包括PH曲线偏移的修剪程序; PH曲线的实时CNC插值器; PH曲线的最佳形状设计; C2 PH样条的有效构造算法; PH空间曲线的替代代数公式。 ***

Pythagorean-hodograph (PH) curves offer many important computational advantages over the "ordinary" Bezier/B-spline representations commonly used in CAD/CAM applications. Among the attributes that distinguish PH curves from the latter are: exact rectification in terms of a polynomial arc-length function; exact rational representations of their offsets; a closed-form expression, suitable for purposes of optimization, for their elastic bending energies; and a propensity to yield fairer loci --- exhibiting a smoother curver distribution --- in the interpolation of discrete data. Moreover, existing CAD systems can readily take advantage of these properties, since the PH curves are entirely compatible with the Bezier/B-spline representation. This project investigates outstanding theoretical and algorithmic issues concerning the construction and manipulation of PH curves. Their solution will permit the full potential of PH curves to be realized in applications such as rapid prototyping and NC milling. These issues encompass trimming procedures for PH curve offsets; real-time CNC interpolators for PH curves; optimal shape design with PH curves; algorithms for efficient construction of C2 PH splines; and alternate algebraic formulations for PH space curves. ***