This paper explores an efficient Lagrangian approach for evolving point cloud data on smooth manifolds. In this preliminary study, we focus on analyzing plane curves, and our ultimate goal is to provide an alternative to the conventional radial basis function (RBF) approach for manifolds in higher dimensions. In particular, we use the B-Spline as the basis function for all local interpolations. Just like RBF and other smooth basis functions, B-Splines enable the approximation of geometric features such as normal vectors and curvature. Once properly set up, the advantage of using B-Splines is that their coefficients carry geometric meanings. This allows the coefficients to be manipulated like points, facilitates rapid updates of the interpolant, and eliminates the need for frequent re-interpolation. Consequently, the removal and insertion of point cloud data become seamless processes, particularly advantageous in regions experiencing significant fluctuations in point density. The numerical results demonstrate the convergence of geometric quantities and the effectiveness of our approach. Finally, we show simulations of curvature flows whose speeds depend on the solutions of coupled reaction--diffusion systems for pattern formation.

翻译：本文探讨了一种用于光滑流形上点云数据演化的高效拉格朗日方法。在此初步研究中，我们聚焦于平面曲线的分析，最终目标是为高维流形提供传统径向基函数（RBF）方法的替代方案。具体而言，我们采用B样条作为所有局部插值的基函数。与RBF及其他光滑基函数类似，B样条能够逼近法向量、曲率等几何特征。当正确建立后，使用B样条的优势在于其系数具有几何意义：这些系数可像点一样进行操控，便于快速更新插值函数，且无需频繁重新插值。因此，点云数据的删除与插入成为无缝过程，在点密度波动显著的区域尤其具有优势。数值结果验证了几何量的收敛性及本方法的有效性。最后，我们展示了曲率流的模拟案例，其演化速度取决于用于模式生成的耦合反应-扩散系统的解。