空间凸三角网上的光滑二次插值

二次曲面（锥面、椭球面、抛物面等）是人们最熟悉的曲面类型,它在飞机机身的外形设计及体素拼合等方面都有广泛应用,但用二次曲面来插值一组给定的空间散乱数据却比较少见。本文提出一种空间凸三角网上构造分片光滑插值曲面的方法,它具有所需顶点信息少,计算方便等优点。

SMOOTH QUADRATIC INTERPOLATION OVER A CONVEX 3D TRIANGULATION

It is an important topic to construct piecewise smooth surface interpolating to 3D scatted data not only in CAGD but also in finite element analysis and other fields. Although quadratic surface（cones,ellipsoids, etc. ）has been widely used in the body shape design of airplane and CSG,it has got very little use in the interpolation to 3D scatted data.The main reason is that the quadratic surface has strong rigidity which makes it difficult to piece two quadratic surface patches together with even only C1 continuity.In this paper, a method of constructing piecewise quadratic surface interpolating to given 3D scatted data with VC1 continuity is presented.The methods based on a theorem given in the paper,which shows how many conditions are required to guarantee that two quadratic surface patches have VC1 cpntinuity on a plane. At first,for the given 3D data （xi,yi,zi,）, a convex triangulation is constructed （only this case is considered） and a normal vector at each vertex is estimated. Then, a quadratic surface patch is defined over each space triangle, which satisfies the interpolating conditions at the three vertexes of the triangle. Because any two adjacent space triangles are uncoplane,there forms a "gap" between these two surface patches.The third step, three quadratic surface patches which are jointed with VC1 continuity are constructed to fill the "gap" .The fourth step, suitable boundary conditions are given to get reasonable continual equations. Finally, we obtain the required surface expressions by solving the continual equations and finding the parameter regions.This kind of surface needs a little information at each vertex and can becalculated conveniently.