保几何特征的渐近迭代逼近造型方法及其应用

Progressive iteration approximation, which shares a simple iteration formula and can avoid solving equations, has achieved rapid development in recent years and has been widely applied to the geometric modeling in the fields of computer aided design and digital geometric processing. Based on important geometric features of curves and surfaces, this project studies geometric feature preserving modeling methods by progressive iteration approximation and its applications. It will be studied in the following directions: progressive iteration approximation methods for regular ordered data; progressive iteration approximation methods by B-splines for point clouds and triangle meshes; deformation and editing of B-splines by progressive iteration approximation methods; the construction of geometric partial differential equations with good properties. By the tools of B-splines and partial differential equations, the project will study many geometric processing problems of point clouds and triangular meshes, such as approximation, deformation and editing, so as to establish the methods and theory for geometric modeling. For this goal, it will discuss the convergence and stability of the progressive iteration, design efficient and fast algorithms, and analyze the accuracy for approximation. In the areas of CAD/CAM, computer graphics and computer animation, many application problems can be solved in a unified framework: the practical problem is firstly converted to a progressive iteration approximation problem with certain geometric feature required to be satisfied, then it can resort to the proposed progressive iteration approximation for the numerical solution and theoretical analysis.

渐近迭代逼近方法由于其迭代简单和无需求解方程组在近几年来得到迅速发展，并应用于计算机辅助设计和数字几何处理等领域的几何造型中。本项目从曲线和曲面的重要几何特征出发，研究保几何特征的渐近迭代逼近造型方法及其在相关领域的应用。具体内容包括：有序规则型数据的渐近迭代逼近方法；点云数据和三角网格的B 样条渐近迭代逼近方法；B 样条变形和编辑的渐近迭代逼近方法；具有良好性质的几何偏微分方程的构造。拟以B样条和偏微分方程为工具，利用保几何特征的渐近迭代逼近方法，研究点云数据和三角网格的逼近、变形和编辑等几何处理问题，建立几何造型的方法和理论。讨论渐近迭代的收敛性与稳定性，设计高效快速的算法，分析误差精度。在计算机辅助设计与制造、计算机图形学以及动画处理等相关领域，将其中的许多应用问题先转换成在满足一定几何特征要求下的渐近迭代逼近问题，然后进行统一处理并开展数值求解和理论分析。

渐近迭代逼近方法由于其迭代简单和无需求解方程组在近几年得到迅速发展，并应用于计算机辅助设计和数字几何处理等领域的几何造型中。本项目从曲线和曲面的重要几何特征出发，研究保几何特征的渐近迭代逼近造型方法及其在相关领域的应用，讨论渐近迭代的收敛性与稳定性，设计高效快速的算法，分析误差精度。主要从五个方面开展项目的研究内容：渐近迭代逼近理论，曲线拼接，能量优化，多项式逼近，偏微分方程数值解。主要成果包括：提出B样条拟合曲线和曲面的渐近迭代构造方法；提出多条Bezier曲线拼接的multiwise merging算法；提出具有良好几何性质（如strain能，曲率分布）的五次插值曲线的迭代构造方法；提出Bezier曲线降阶的新方法，以及有理Bezier曲线的多项式逼近方法；提出偏微分方程基于三角Bezier曲面的数值解的迭代方法。通过本项目的研究，形成了保几何特征的渐近迭代逼近造型方法的理论框架，为CAD技术提供新的几何造型方法。