CAD中PH曲线的几何理论及其应用研究

项目名称： CAD中PH曲线的几何理论及其应用研究

项目编号： No.61272300

项目类型： 面上项目

立项/批准年度： 2013

项目学科： 自动化技术、计算机技术

项目作者： 汪国昭

作者单位： 浙江大学

项目金额： 80万元

中文摘要： 本项目研究PH等曲线的几何理论，发掘它们特有的几何特征，并应用于CAD领域。 PH曲线优点突出，应用广泛。但目前仅三次、四次和五次本原这三种PH曲线的几何特征为人所知。换言之，在应用中，只有这三种曲线的几何手段得到充分发挥；除此以外，对Bezier曲线行之有效的几何手段在PH曲线的范围内都受到限制，满足不了实际需要。本项目要改变此局面，使受限制的几何手段对PH曲线同样起作用。 PH曲线的几何理论是CAGD的难题，长期以来未找到解决方法。近年，有了突破。本项目是原创性研究，要创造新的研究方法，发现上述三种以外的PH曲线的本质属性在控制多边形上的反映，以边长与角度刻画之。从而建立起直观性强、操作方便、表示简洁的几何理论，使得运用几何手段可以对PH曲线作判别，可以在PH曲线范围内作交互设计，像使用Bezier曲线一样方便、灵活、有效。再深入地研究OR曲线以及它们生成的样条等曲线的几何理论。

中文关键词： PH 曲线；OR曲线；几何特征；控制多边形；极小极小曲面

英文摘要： In this project, we are proposed to study the theory of PH curves geometrically, finding their geometrical characteristics and making them apply to CAD field. PH curves have been used widely for their strong advantage. Until now only 3 types of PH curves have been studied well on their geometrical theories, which are cubic, quartic and original quintic PH curves. Therefore, some limitations exist when we want to use analogous effective geometrical approach for PH-curves as what we did frequently for Bézier curves. We will overcome these bottlenecks and strengthen the geometrical approach suitable for PH curves. Basically, the study of geometrical characteristics for PH curves is troublesome, no one has found any effective breakthrough of it. However, as an original research work, we have found a approach to describe the essential attributes of PH curves using the edge length and angle of their control polygons. This approach can deduce many essential geometrical results intuitively. Based on these results, we can design and operate PH curves interactively, just as flexible as the Bézier curves. Furthermore, we will make further research on OR curves and corresponding geometrical theory of the splines they generated.

英文关键词： PH curve；OR curve；geometrical characterization；control polygon；minimal surface