关键词: 对数映射, 法坐标, 广义镜像, 扩展德布尔算法, 重用

Abstract: 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 B­spline 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 curves 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.

Key words: Generalization of de boor algorithm, Generalized mirror symmetry, Logarithmic mapping, Normal coordinates, Reuse