计算数学

    摘 要 NURBS方法在计算机辅助几何设计（CAGD）中得到广泛的应用，参数定义在矩形区域上的非均匀有理B样条曲面（NURBS）与参数定义在三角区域上的仿射坐标系下的曲面是造型系统的主要工具，其相应的deBoor递推过程是计算机实现的主要方法。本文以计算机辅助几何设计中的NURBS方法为研究对象，对deBoor计算过程进行了误差估计，对样条曲面之间的几何连续性和光滑拼接做了一定的分析和研究，文章的后半部分着重论述了样条方法在图像处理、曲面造型，以及计算可视化等方面的应用实例和开发成果。全文共分五章，主要内容如下： 文章首先介绍了NURBS乘积型曲面，参数定义在三角域上的B-patch，以及区间与三角域上的deBoor递推过程。考虑了当控制点存在扰动时对计算结果的影响，并对计算中产生的舍入误差进行了分析，并得到了其误差估计式。 单片NURBS曲面具有较好的参数与几何连续性。而在实际造型系统中，经常需要将不同的曲面片加以拼接。我们利用G连续的充分条件以及B样条基函数的导数性质，构造了具有q阶公共边界的NURBS曲面之间实现G1（切平面连续）与G2（高斯曲率连续）光滑拼接的实用算法。即根据一张已知的NURBS曲面片，通过调整边界附近的部分控制点和权因子，以达到光滑拼接的目的。 计算机辅助几何设计具有广泛的应用前景。利用NURBS方法及曲面造型技术，我们开发了一系列应用软件和开发工具，主要包括： 在图象处理中，我们利用样条曲面设计滤波模板，取得了较大的灵活性，在指纹图像自动识别系统中得到了成功的应用。 利用NURBS方法，我们在VC++环境下开发了自由参数曲面和隐式曲面的绘制算法及图形生成工具，具有通用性和易用性。 在计算可视化方面，我们在数值计算所获得的大量数据的基础上，开发了波动方程和热扩散方程解的三维图形动画，取得了较好的计算机视觉效果。 [英文摘要]：     Abstract The NURBS method has been widely used in CAGD(Computer Aided Geometry Design), parametric piecewise national surfaces based on tensor product surfaces and surfaces over triangular regions are the main tools for computer modeling design. In CAGD we mainly use the deBoor algorithm to calculate the points on curves and surfaces. This paper take the NURBS method as the research object, make an error analysis on deBoor algorithm, and discuss the continuity on NURBS surfaces. The latter part of this paper takes an emphasis on the application in the field of image processing, modeling and computational visualization. There are totally five chapters, which include: After a brief introduction to the tensor product NURBS , the B-patch, and the deBoor Algorithm, we will discuss how the error in control points affects the final result. Also we make an analysis on the rounding error in the process of deBoor. Non-Uniform Rational B-Splines (NURBS) provides a powerful tool in geometric modeling. Although it is continuous in one single NURBS, in modeling system it is always needed to adjoin two different patches together. With the sufficient condition of G continuous and the properties of the derivative of the base function of B-Splines, this paper provides a practical method to make the adjacent NURBS patches with a boundary of degree q G1(tangent plane )and G2(Gauss curative) continuity. That is, for a given patch we modify the control points and the relative weights of another to make them G1/G2 continuous. CAGD is a very promising field and can be used widely. With NURBS method and the technology of modeling design, we developed some applications, which include: In image processing, we use NURBS to design the filter, and got much more flexibility. In the system of recognizing fingerprint automatically, the filter we designed gives a good effect. In VC++ environment we construct a system for designing the freedom surfaces, and design an algorithm to draw the shape of the implicit function. We make it a convenient tool which is easy to use. In the aspect of computational visualization, based on the data of the numerical computation, we developed the 3-D animation of the solutions to wave equation and diffusion equation, and got a vivid vision on computer.