常见NURBS曲面上的C^2曲线插值法

Interpolation Curve Method with Continuous C^2 on Common NURBS Surface

研究限制在光滑曲面Σ上的曲线插值问题。基于构造一个 D R2 → R3,且像集为Σ的映射 ,将曲面上曲线插值归结为通常 R2上的插值曲线构造 ,避免了一般曲面上测地距离不易求得的难题 ,获得了如球面、圆柱面、圆锥面等常用曲面上的 NURBS插值曲线。进一步用 NURBS构造曲面Σ,解其反演方程 ,在平面上进行NURBS插值 ,给出一般参数曲面 Σ上 C2 曲线插值方法。并对大量实例进行计算 ,分析误差 ,绘出曲线插值图 ,从 RMSE(Root-mean-square error)误差及绘制的实例图看 ,插值效果好。

The interpolation with a curve restricted on smooth surface Σ is concerned. It is a very difficult problem to get the geodesis distance on common surface. When the image set is Σ , based on constructing the mapping: DR 2→R 3 , it turns the origin problem into the general interpolation curve construction on R 2 . The interpolation curve method with C 2 continuous on general surface Σ is given by constructing Σ using NURBS, solving its inverse equation and performing NURBS interpolation on plane. Thus the NURBS interpolation curve on spherical, circular cylindrical and circular conical surfaces, etc is achieved. Furthermore the error analysis and the distribution are given with some examples. From the illustrations of interpolation curve and root mean square error, this method is simple and efficient.