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

毕达哥拉斯-hodograph （PH）曲线与CAD/CAM应用中常用的“普通”Bezier/ b样条表示相比，提供了许多重要的计算优势。将PH曲线与后者区分开来的属性包括：根据多项式弧长函数进行精确整流；对其偏移量的精确理性表示；它们的弹性弯曲能的一个适合于优化的封闭表达式；在离散数据的插值中，倾向于产生更公平的轨迹——表现出更平滑的曲线分布。此外，现有的CAD系统可以很容易地利用这些特性，因为PH曲线与Bezier/ b样条表示完全兼容。该项目研究了有关PH曲线的构建和操作的突出理论和算法问题。他们的解决方案将使PH曲线的全部潜力在快速成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；实时数控PH曲线插补器；PH曲线优化形状设计；C2 PH样条的高效构造算法；以及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. ***