网格曲面上的样条曲线阵列

提出一种流形网格曲面上曲线阵列复制方法，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，使对曲线的操作可以转换为对曲线控制顶点的操作；引入离散指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，按照曲线阵列前后控制顶点法坐标保持不变的原则，建立曲线阵列前后控制顶点的对应关系，实现曲线的有规律多重复制。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，因而也保证了曲线阵列重用过程中的形状保持性。将曲线阵列时的位置与形状分开处理，使曲线生成仅与其所在的局部区域有关而与曲面的整体大小无关，既便于保证曲线的形状保持性，也减少了计算量。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。

Pattern of Spline Curves on Meshes

For the purpose of the curves reuse and redesign, an array replication method of curves on manifold triangulation surfaces is proposed. In our method, the curves on mesh surfaces are denoted by geodesic B-splines, which can transform the operation for curves to the operation for control points of curves. By introducing the discrete exponential mapping theory, the control points of the source curves can be mapped into its tangent space, and their normal coordinates can be obtained. Taking advantage of the invariable property of the normal co- ordinates, the corresponding relation between the pre- and post-array control points are established, with which the regular multiple replication of curves can be realized. The normal coordinates can retain the geodesic distance and relative position of the control points, thus ensuring the shape retention of the curves in reuse and array. The position from the shape of the curve array processing is seperated, making sure that the curve generated only related to the local area and ignore the overall size of the surface. What’s more, it is not only easy to ensure the shape retention of the curve, but also reduce the calculation. Experimental results show that, our method is robust, effective, and can meet the requirements of interactive design of curves on mesh surfaces.