参数曲面上的插值与混合

如何表示曲面上的曲线,在处理诸如数控加工中的路径设计以及CAD/CAM等领域频繁出现的曲面裁剪问题时显得日益重要.给出了数据点的切方向(切方向及曲率向量或测地曲率值)指定而G¹连续(G²连续)插值曲面上任意点列的方法.作为曲面上曲线插值问题的特例,还讨论了曲面上曲线的混合问题.基本思想是借助于微分几何的有关结论,曲面上曲线的插值问题被转化为其参数平面上类似的曲线插值问题.该方法能够用二维隐式方程来表示曲面上的插值曲线,从而把在显示该曲线时所面对的曲面求交的几何问题转化为计算隐式曲线的代数问题.实验证明该方法是可行的,而且适用于CAD/CAM及计算机图形学等领域.

Interpolation and Blending on Parametric Surfaces

Representing a curve contained in a surface is very important in dealing with path generation in computer numerical control (CNC) machining and the trimming issues that frequently occur in the field of CAD/CAM. This paper develops methods for tangent direction continuous (G¹) and both tangent direction and curvature continuous (G²) interpolation of a range of points on surface with specified tangent and either a curvature vector or a geodesic curvature at every point. As a special case of the interpolation, the blending problems of curves on surface are also discussed. The basic idea is as follows: with the help of the related results of differential geometry, the problem of interpolating curve on a parametric surface is converted to a similar one on its parametric plane. The methods can express the G¹ and G² interpolation curve of an arbitrary sequence of points on a parametric surface in a 2D implicit form, which transforms the geometric problem of surface intersection, usually a troublesome issue, into the algebraic problem of computing an implicit curve in displaying such an interpolation curve. Experimental results show the presented methods are feasible and applicable to CAD/CAM and Computer Graphics.