曲面造型中散乱数据插值曲面问题的研究

【作者】 高珊珊；

【导师】 张彩明；

【作者基本信息】 山东大学 ， 计算机应用技术， 2005， 硕士

【摘要】 本文对曲面造型中散乱数据插值曲面问题进行了研究。构造散乱空间数据插值曲面技术在CAD、计算机图形学、气象和勘探等各类科学研究和工程设计中有广泛的应用。 由于工程曲面的不规则性,以及散乱数据的无明确规律和无序性,很难用单一的数学形式把曲面表达出来,因此一般采用分片的方法设计曲面,最后将各曲面片光滑的连接起来,形成一个完整的曲面。目前最常用的曲面片是三边曲面片和四边曲面片。由于三角插值方法的几何意义明显,便于调整,成为重要的曲面构造方法。构造插值曲面的常用方法之一就是对给定的散乱数据进行三角剖分,根据边界连续条件构造每个三角形区域上的插值曲面片,整体的插值曲面由各个曲面片拼合而成。三角域上的插值包括有理插值和多项式插值两种方法,其中多项式插值因为其结构简单,易于计算的优点,应用尤为广泛。 不同的插值方法常会有各自的缺点,大致会有以下几种:构造方法复杂,所需要的插值条件限制较多;插值方法限制给定数据点个数不是太大时是可行的;对所形成的三角形网格的顶点的某些特性进行限制;在数据点的局部区域上形成一个整体曲面,而整体曲面有时很难具有数据点所建议的曲面形状,或者所构造的整体曲面局部调整性较差;由运算结果得不到唯一的插值曲面等。 针对上述问题,该论文提出了一种构造插值曲面的新方法。新方法把空间数据点作三角划分,在每个顶点处构造一个二次分片多项式曲面片,每个三角形上的曲面片由三个顶点处的曲面片加权平均产生。由于在每个顶点处构造的是C~1连续的二次分片多项式曲面片,所以用新方法构造的插值曲面具有较好的保形性和局部调整性。此外,文章详细分析了由边界连续条件构造的方程组在不同情况下的求解过程,给出了简单方便的求解方法。新方法可有效地构造对空间散乱数据点进行插值的光滑曲面。所构造的插值曲面具有二次多项式的插值精度。 考虑到不同的应用背景,对构造光顺曲面,文章又提出了一种新的基于能量最小准则的散乱数据点多项式插值方法。构造局部曲面片所需的求解条件仅从该区域上获得,根据能量最小准则确定未知量,使构造的曲面具有更为理想的局部调整性。新方法所构造的曲面具有原始数据点所建议的形状,并且对原数据点褶更多还原

【Abstract】 The work studies the problem of interpolation to scatter data points in surface modeling. The technique of constructing interpolant to scatter data points is used widely in many fields of scientific research and engineering design, such as CAD, the computer graphics, the meteorology and the exploration and so on.As irregularity of surface in project application, and as randomness and disorder of scatter data points, it’s difficult to express the surface with a sole mathematical formalism. Therefore the piecewise method is used generally in the surface design, finally all the surface pieces are connected smoothly to form a whole surface. The most commonly used surface patches are the triangle and the quadrangle surface patch. As the triangular interpolation method has obvious geometry significance, and the surface constructed is easy for the adjustment, more and more reseachers pay their attention to the method. At present one of commonly used methods of constructing interpolated surface can be described simply as follows: the given scattered data points are triangulated into triangle network, the surface patch over each triangle region is formed according to boundary condition of continuity, and each surface patch joins together to form overall interpolated surface with C~1 continuities. There are two methods for interpolation on triangle network, one is rational interpolation, another polynomial interpolation. Because of the merit of simple structure and easy calculating, polynomial interpolation is used more widely. Now there are many methods to construct polynomial surface to the scattered data points.Different interpolation methods often have the different shortcoming, and it can be classified simply as follow: The process of constructing is too complex and the interpolation condition is limited by many factors; The method for interpolation is feasible just at the condition of that the number of the given scattered data points is not too big; The triangle grid vertex must has certain characteristics to satisfy the need of constructing; Just forms an overall surface over the whole partial region of each data point, and the overall surface sometimes is difficult to possess the surface shape suggested by the given data points, or can’t be adjusted easily; Cannot obtain the onlyinterpolated surface by the operation result and so on.In view of above question, this article presents a new method to construct interpolated surface to the scattered data points. New method triangulates the given data points into triangle network, and at the adjacent region of each point a C1 piecewise quadric interpolation patch is constructed. The surface patch on each triangle is constructed by the weighted combination of the three quadric patches at the vertices of the triangle. All the triangle patches are put together to form the whole surface with C1 continuities. Because the surface patch at the adjacent region of each point is C1 piecewise quadric interpolation patch, it has the property of keeping shape and can adjust the shape easily over the local region. In addition, the solution process of equations formed by boundary condition of continuity in different situation is analyzed, and the simple and convenience solution is given. The new method can construct smooth surface interpolating the given scattered data points effectively, and the polynomial precision set of the method includes all the polynomials of degree two.Considering different background of application, for constructing the fairing surface, a new method of interpolation based on the criterion of minimum energy is presented. The solution conditions used to construct partial surface is only obtains from this region, the unknown parameter is acquired according to the criterion of minimum energy, so it enable the constructed surface to have a more ideal partial adjustment. The new surface has the shape suggested by the primitive data points, and is more fairing over the region with bigger drape. At last examples are given to make comparison with previous method.But the above two methods finally constructs the polynomial interpolations of degree seven, which is higher for C1 interpolation with polynomial. In the following research we will make the improvement to the weight function by reduces its degree in order to obtain lower degree polynomial interpolation. The fairness of surface is an extremely important research question in CAGD, the fairness of surface can be improved based on criterions of the fairness in future research.更多还原