网格曲面上测地B样条曲线交互操作与重用

针对现有曲面上自由曲线设计重用方法的不足,提出一种流形网格曲面上曲线几何变换方法,达到曲线重用与再设计的目的。网格曲面上的曲线用测地B样条表示,具有与欧氏空间中传统B样条相一致的明确数学模型;引入对数映射理论将给定的源曲线控制顶点映射到切空间,获得它们的法坐标,按照曲线迁移前后控制顶点法坐标保持不变的原则,建立曲线迁移前后控制顶点的对应关系,实现类似于欧氏空间中的平移、旋转和缩放等几何变换。以网格曲面上离散对数映射理论为基础,将欧氏空间中的对称定义拓展到曲面空间,提出曲面上曲线的广义镜像概念并给出具体的算法实现。法坐标很好地保持了控制顶点之间的测地距离和相对位置关系,因而也保证了曲线迁移重用过程中的形状保持性。试验结果表明,所介绍方法健壮、有效,能满足曲面上曲线的交互设计要求。

Interactive Manipulation and Reuse of Geodesic B-spline Curves on Meshes

In allusion to the deficiencies of the existing methods of reuse designing free curve on the surface,a geometric transformation method of curves on Manifold triangulation surface is proposed to achieve the aim of curves reuse and redesign.The curve on the mesh surface is represented as Geodesic B-spline curve,which has the clear uniform mathematical model with the classical B-spline curves in Euclidean space;by introduction of the logarithmic mapping theory,the control points of source curves can be mapped into tangent space and its Normal Coordinates can be obtained.According to the principle of those Normal Coordinates of remained unchanged,establishing the corresponding relation between pre and post transfer of curves,and curve’s translation,rotation and scaling could be realized similaring to its geometric transformation in Euclidean space.The symmetry definition in Euclidean space is expand to curved space based on discrete logarithmic mapping theory,the generalized mirror symmetry concept of curve on the surface is proposed and its algorithm implementation is given.The Geodesic distance and relative position of those control points can be nicely maintained by using the Normal Coordinates,and it ensures the shape preserving property during transfer and reuse of curves.The results show that the method is robust,effective,and able to meet the requirements of curve interaction design on mesh surface.