插值有理Bézier渐近四边形的有理Bézier曲面

摘要

为构造封闭的曲线为有理Bézier曲面的边界渐近线,给出封闭四边曲线为渐近四边形的条件,并提出插值该四边形的曲面构造方法.首先在给定角点数据的前提下构造优化的n次有理Bézier渐近四边形;然后利用该四边形和曲面在四边形上的切矢确定曲面沿边界的两排控制顶点和权;最后极小化曲面薄板能量函数确定剩余自由的控制顶点,进而构造出光滑的双5n-7次有理Bézier插值曲面.实例展示边界曲线为有理3,4,5次时曲面的构造结果,以及边界曲线含有直线或者拐点的情况,表明该方法是可行的.%Conditions for construction of rational Bézier surface interpolating a closed quadrilateral as its asymptotic boundary are presented.Firstly,from the given comer data,an optimized rational Bézier asymptotic quadrilateral of degree n is constructed.Secondly,two arrays of control points and weights along the boundary curves are obtained from the quadrilateral and the tangent vectors of the surface.Finally,minimizing the plate spline energy determines the other free control points and then a smooth rational Bézier surface ofbi-(5n-7) degree is constructed.Some representative examples show the construction of surfaces interpolating the cubic,quartic or quintic rational Bézier asymptotic quadrilaterals and verify the effectiveness of the method.