目的 样条曲线曲面的构造是工程制图中的一个重要部分。针对双曲抛物面上参数样条曲线的构造,在已有的研究基础上提出了一种样条方法使曲线曲面可以任意地逼近一个多边形或者一个网格。方法 在标准四面体内构造一个双曲抛物面,在该曲面上以基函数参数化的方法定义一种带形状参数的参数样条曲线曲面,样条基函数通过将双曲抛物面的有理参数化进行限定,生成单参数有理样条基函数。详细研究了样条的保形性及其端点性质。结果 样条曲线具有一个可变的形状控制因子,可以对曲线进行调整,能以任意精度逼近这个控制四边形或网格。对空间节点列,利用该样条可以生成G²-连续空间曲线,同样对于空间网格可以构造G²-连续的拟合曲面,它所对应的基函数可以是有理形式。结论 实验结果表明,本文在笔者已有的研究基础上提出的参数样条曲线可以通过重心坐标系变换适应为任意的四边形,除了空间四面体内的样条曲线,四面体退化成四边形同样可实现。

Objective Spline curve surface is an important part of CAGD. To obtain the parameter spline curve on the hyperbolic paraboloid, the four rational basis functions presented in this paper have a changeable control handle and a unary function. Method A hyperbolic paraboloid is constructed in the standard tetrahedron. On the surface, the base function defines a parameterized method with the parameters of the shape parameter spline curve surface. The basis function of spline through the rational parameters of the hyperbolic paraboloid is limited. A single-parameter rational basis function of spline is generated. Additionally, the nature of shape preserving and endpoints are considered. Result The nature of shape preserving and endpoints are considered. The curvature of two endpoints are derived as zero for a range of points. Consequently, a G²-continuity curve can be attached by this spline curve without any additional requirements. Conclusion Experiment shows that we use this spline to raise the surface for spatial mesh, which also can implement G²-continuity with much freedom.