Advanced Construction, Analysis, and Application Algorithms for Pythagorean-Hodograph Curves

毕达哥拉斯hadograph （PH）曲线是一种特别通用的自由形式参数曲线，在基于几何的应用中，如计算机图形学、计算机辅助设计、CNC加工、快速原型设计、运动设计、形状分析和重建，它提供了比“普通”贝塞尔/ b样条曲线更独特的计算优势。PH曲线最初是为解析几何的两个基本问题提供精确的解决方案：计算弧长和自由曲线的偏移量。然而，最近变得明显的是，它们为更广泛的问题提供了严格的算法，包括从中轴变换中精确重建形状的边界，用于CNC加工和机器人轨迹生成的精确和通用的实时插值算法，基于精确（非线性）能量最小化的“公平”形状的构建和分析，用于计算机动画的旋转最小化帧的构建等。这些算法采用了一系列复杂的数学概念，包括平面PH曲线的复杂变量表示，中轴变换的Minkowski-metric PH曲线，傅里叶曲率分析的轮廓积分，PH样条构造的同伦方法等。该项目将研究平面和空间PH曲线的构建、分析和应用中的突出理论和算法问题，其解决方案将使其在实际应用中充分发挥潜力。这项研究将与工业合作者合作进行，以便根据PH曲线与CAD系统的标准Bezier/ b样条表示的兼容性，促进技术的快速转让。

The Pythagorean-hadograph (PH) curves are an exceptionally versatile class of free-form parametric curves, that offer unique computational advantages over "ordinary" Bezier/B-spline curves in geometry-based applications such as computer graphics, computer-aided design, CNC machining, rapid prototyping, motion design, and shape analysis and reconstruction. The PH curves were originally introduced to provide exact solutions to two basic problems of analytic geometry: the computation of arc lengths and offsets for free-form curves. It has recently become apparent, however, that they offer rigorous algorithms for a much broader range of problems, including exact boundary reconstruction of shapes from medial axis transforms, exact and versatile real-time interpolation algorithms for CNC machining and robot trajectory generation, construction and analysis of "fair" shapes based on precise (non-linear) energy minimization, construction of rotation-minimizing frames for computer animation, etc. These algorithms employ a sophisticated array of mathematical concepts, including complex variable representations for planar PH curves, Minkowski-metric PH curves for medial axis transforms, contour integration for Fourier curvature analysis, homotopy methods for PH spline constructions, etc. This project will investigate outstanding theoretical and algorithmic problems in the construction, analysis, and applications of planar and spatial PH curves, whose solution will permit their potential to be fully realized in practical use. The research will be conducted in cooperation with industrial collaborators, to facilitate a rapid transfer of the technology based on the compatibility of PH curves with the standard Bezier/B-splime representations of CAD systems.