具有分片有理等距面的自由曲面造型方法

Freeform surface modeling with some geometric constraints is an important problem in a lot of applications. We study the problem of freeform surface modeling using surface patches from Dupin cyclide which is called the cyclide spline surface. By relaxing the vertex positions in the supporting mesh of a cyclide spline surface, we employ a global optimization method for computing a cyclide spline surface. In contrast to all existing approaches to this problem which are based on a strategy of vertex frame propagation, our approach treats all vertex positions and all surface frames attached to vertices as variables in a minimization problem. Using this method, we have a global control on the surface shape as well as the fairness of a surface. .The two problems we consider are cyclide spline surface fitting and interactive shape design and modification with cyclide spline surfaces. We study some techniques for obtaining a cyclide spline surface fitting to a reference surface with high quality. We study the approximation degree of a cyclide spline surface. We employ a strategy of normal interpolation as an additional constraint to improve the approximation accuracy of a cyclide spline surface. We study the rules of local refinement,the handling of T-joints and umbilical points. We also propose methods to improve surface fairness, including a method for controlling the curvature variation of some streamlines on a cyclide spline surface, as well as employing the Willmore energy for fairness control. We study the shape design of a cyclide spline surface using a rough control mesh and shape modification of a cyclide spline surface by editing some points or other shape features on a cyclide spline surface. . By proposing solutions to a series of key problems in cyclide spline surface modeling, we provide new techniques for a freeform surface design and editing system using cyclide splines surfaces.

带几何约束的自由曲面造型问题有着重要的应用背景。本项目研究具有精确分片有理等距面的自由曲面造型问题。我们使用Dupin四次圆纹曲面片拼接构成 G1连续的自由曲面，我们称这种曲面为圆纹样条曲面。Dupin四次圆纹曲面片的等距面具有理二次形式，与CAD领域的行业标准NURBS形式兼容；圆纹样条曲面由圆弧构成，曲面上的曲率线是圆弧样条曲线。以前的方法无法构造具有复杂形状的圆纹样条曲面。我们提出整体优化的方法：通过允许顶点位置移动，增加曲面表达的自由度；把曲面支撑网格的所有顶点和顶点上的标架作为优化的变量，对曲面的整体形状进行控制，构造出形状复杂的圆纹样条曲面。在该整体优化的框架下，我们针对高质量的圆纹样条曲面的拟合和圆纹样条曲面的交互设计和修改这两方面，研究其中的一些关键问题，为圆纹样条曲面造型系统提供技术和理论支持。

本项目研究了基于特殊曲面片拼接的自由曲面建模技术，主要研究基于圆纹曲面片拼接的复杂曲面造型和复杂可展曲面的交互建模。主要研究成果有：1）提出了有效的圆纹样条曲面设计技术和拟合技术。主要创新包括提出了基于曲面片拼接的曲面造型整体优化框架、圆纹曲面的局部细分优化技术和利用球面处理曲面上脐点的技术。2）提出了可展B样条曲面的快速交互建模技术和快速拟合技术。主要创新包括将可展B样条建模问题表述为具有线性和二次约束的优化问题、快速的优化求解技术和考虑曲面法向的快速稳定的可展B样条曲面拟合算法。相关研究成果被图形学领域顶级期刊ACM Transactions on Graphics (TOG) 和 Computer Aided Geometric Design (GMP 2016特刊) 录用。..圆纹曲面和可展曲面的特殊几何性质与应用领域密切相关，对设计高效的曲面数控加工算法、设计具有高度相似性节点的建筑表面支撑结构、产品物料的最优裁剪和展开等有重要意义。我们提出的建模技术解决了圆纹曲面自由度太少、难以构造复杂形状的问题，开拓了圆纹曲面在计算机辅助几何设计领域的应用前景；我们提出的可展曲面交互建模算法使得构造具有复杂形状的可展曲面模型变得简单直观。鉴于可展曲面在船体造型建筑表面设计和服装造型等领域的应用，交互可展曲面建模技术将对相关产品的设计和制造带来帮助。总之，本项目的研究成果对提高产品加工的效率和减少生产成本等方面有重要价值。

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