Title: Meshing of High-Dimensional Toroidal Manifolds from Quasi-Periodic Three-Body Problem Dynamics using Parameterization via Discrete One-Forms Show Chinese title

Title: 高维环面流形从准周期三体问题动力学的网格划分，使用离散一形式的参数化

Abstract: High-dimensional visual computer models are poised to revolutionize the space mission design process. The circular restricted three-body problem (CR3BP) gives rise to high-dimensional toroidal manifolds that are of immense interest to mission designers. We present a meshing technique which leverages an embedding-agnostic parameterization to enable topologically accurate modelling and intuitive visualization of toroidal manifolds in arbitrarily high-dimensional embedding spaces. This work describes the extension of a discrete one-form-based toroidal point cloud meshing method to high-dimensional point clouds sampled along quasi-periodic orbital trajectories in the CR3BP. The resulting meshes are enhanced through the application of an embedding-agnostic triangle-sidedness assignment algorithm. This significantly increases the intuitiveness of interpreting the meshes after they are downprojected to 3D for visualization. These models provide novel surface-based representations of high-dimensional topologies which have so far only been shown as points or curves. This success demonstrates the effectiveness of differential geometric methods for characterizing manifolds with complex, high-dimensional embedding spaces, laying the foundation for new models and visualizations of high-dimensional solution spaces for dynamical systems. Such representations promise to enhance the utility of the three-body problem for the visual inspection and design of space mission trajectories by enabling the application of proven computational surface visualization and analysis methods to underlying solution manifolds. Show Chinese abstract

Abstract: 高维视觉计算机模型有望彻底改变航天任务设计过程。 圆形限制性三体问题（CR3BP）产生了高维环面流形，这些流形对任务设计师来说具有极大的兴趣。 我们提出了一种网格划分技术，该技术利用与嵌入无关的参数化方法，以实现任意高维嵌入空间中环面流形的拓扑准确建模和直观可视化。 这项工作描述了基于离散一形式的环面点云网格划分方法扩展到高维点云的应用，这些点云沿CR3BP中的拟周期轨道采样。 通过应用与嵌入无关的三角形方向分配算法，所得网格得到了增强。 这显著提高了在降维投影到三维后解释网格的直观性，以便于可视化。 这些模型为高维拓扑提供了新颖的基于表面的表示方法，而迄今为止这些拓扑仅被展示为点或曲线。 这一成功证明了微分几何方法在表征具有复杂高维嵌入空间的流形方面的有效性，为动力系统高维解空间的新模型和可视化奠定了基础。 此类表示有望通过使已验证的计算表面可视化和分析方法应用于潜在的解流形，从而增强三体问题在航天任务轨迹视觉检查和设计中的实用性。