自由曲面模型全局美化技术

利用可微流形的无穷小变形技术,实现紧公差约束下对点云逼近的多张B-样条曲面的全局美化.根据微分流形上的Beltrami-Laplace算子,定义反映曲面整体形状变化的变形能量泛函,并得到泛函的唯一极小解.给出两张单节点B-样条曲面间G1连续的充要条件以及公共边界曲线控制顶点的本征方程,并利用局部格式构造整体收敛的G1光滑拼接的边界表示模型获得变形映射族的特解.特解的构造不可避免地导致模型在缝合区域的形状产生局部瑕疵,影响反求模型的保形性.通过特解来构造能量泛函的极小解,使得这类似应力集中的效应被逐步松弛到曲面的内部,从而改善模型的整体形状.应用实例揭示自由曲面的全局美化技术在反求工程中的意义.

GLOBAL BEAUTIFICATION TECHNIQUE FOR FREEFORM SURFACE MODEL

The multiple surfaces approximated to point cloud subject to tight error are beautified globally by using the infini- tesimal deformation technique of differentiable manifold. The deformation energy functional reflecting the overall shape of a surface is defined by using the Beltrami-Laplace operator on the manifold. Also the unique solution of the minimum of the en- ergy functional is formulated according to the property of har- monic function. Then, the necessary and sufficient conditions of G1 continuity between two B-spline surfaces with single knots are given and simplified, as well as the intrinsic equations of control points of the common boundary curve. Based on the local scheme of convergent G1 smooth surfaces, a special solu- tion of the family of deformation maps is constructed. The spe- cial solution is represented by the smoothly stitched B-rep model. The inevitable local imperfection at the stitching regions caused by constructing the special solution greatly influences on the shape preservation of reverse engineered model. Finally, the final solution is constructed such that the deformation en- ergy clustering round the stitching regions is released gradually to the surface interior. Consequently, the shape of the model is improved. The practical examples reveal the value of the global beautification technique in reverse engineering.