网格曲面的逆向设计与特征计算研究

Computer Aided Geometric Design (CAGD) is different from computer graphics (CG) in service object, data measurement, technique of expression and theoretic support of technology. CAGD has greater influence on and offer more support for CG since it has more solid foundation of mathematics. The goal of our project is to open a theory channel from CAGD to CG, so that the important research results of our team can be transplanted from CAGD to CG, and taking mesh surface pencil, geodesic distance field of mesh, harmonic operator of mesh, lines of curvature on mesh and gradient bounds of mesh as breakthrough point, to do a research on three kinds of reverse design as well as three kinds of characteristic computations for mesh. To be specific, design of pencil of mesh surface with common discrete geodesic; design of harmonic mesh surface with space closed polygon as its boundary; simplification of mesh preserving discrete geodesic and/or discrete lines of curvature; computation of discrete lines of curvature on mesh; computation of discrete geodesic loop with high precision; and estimation of bounds on gradient of first and second orders. The originality and novelty of this project lie in it is completely contrary to the previous work which is to compute discrete geodesics or discrete difference values of given mesh. And it will provide more wide and more proper mesh choice for users. Meanwhile, to derive the characteristic computation of mesh surfaces is an unprecedente research, and it will reveal the brand-new geometric and algebraic features of mesh surfaces, and hence strengthen the functions of software system observably.

计算机辅助几何设计(CAGD)在服务对象、数据量度、表现手段、理论支撑上不同于计算机图形学(CG), 然而其数学沉淀更为深厚, 可给CG以更多影响与支持. 本申请旨在建立一条从CAGD到CG的理论通道, 把本研究团队近年CAGD重要成果移植到CG, 以网格的曲面束、测地距离场、调和算子、曲率线和梯度界等为突破口，进行网格的3项逆向设计与3项特征计算研究. 具体来说就是: 具有公共离散测地线的网格曲面束设计, 以空间闭多边形为边界的调和网格曲面设计,保离散测地线与(或)保离散曲率线的网格曲面简化, 网格曲面的离散曲率线计算, 离散测地环高精度计算, 以及1,2阶梯度界估计. 本项目的独创新颖性在于,它与以往已知网格求离散测地线或离散差分值的操作走向完全相反, 将给用户提供更广泛与更合适的网格选择; 而网格曲面的这些特征计算则前所未有, 将揭示网格崭新的几何及代数特征,显著地强化软件系统功能.

本项目旨在建立一条从CAGD（计算机辅助几何设计）到CG（计算机图形学）的理论通道, 把本研究团队与合作单位的CAGD重要成果移植到CG. 近4年以来，我们以自己的研究成果为起点，瞄准“以已知曲线为公共特征线的曲面束”这个国际前沿热点作积极推广及持久攻关，在带有离散测地线/曲率线的网格曲面束、带有B样条曲率线/测地线的NURBS曲面束、NURBS曲面导矢界，以及网格补洞/求交/变形、网格误差的区间分析、渐进迭代逼近、细分、曲线重构/参数化等离散型或连续型曲面的特征计算的理论研究与实际应用中取得了一批崭新成果，推导了一系列定理与算法，为CAD与CG提供了高质高效的工具，顺利完成了预定计划...本研究共发表1篇SCI论文，10篇EI论文，1本专著，1本专著的完整1章，2篇国际国内会议论文，以及其它10篇国内外期刊论文；其中2篇论文获得最佳学生论文奖，1人获几何设计与计算杰出贡献奖；培养毕业硕士生8人...本项目的独创新颖性在于，它与以往已知网格求离散测地线或离散差分值的操作走向完全相反，将给用户提供更广泛与更合适的网格选择；而网格曲面或NURBS曲面的这些特征计算则前所未有，将揭示相应曲面的崭新的几何及代数特征，显著地强化软件系统功能.