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

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.