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

TL;DR

The paper addresses the challenge of visualizing high-dimensional toroidal invariant manifolds arising in the circular restricted three-body problem (CR3BP). It introduces an embedding-agnostic meshing approach based on discrete one-forms, k-nearest-neighbors graphs, and the minimum cycle basis to obtain a 2D parameterization on surface patches, enabling Delaunay triangulation in the parameter space and meshing in high-dimensional embedding spaces. The authors demonstrate, for the first time, a mesh of a 4D toroidal manifold sampled by a standard map and a meshed invariant manifold in 6D state space (Enceladus), with downprojection to 3D and orientation-based coloring to reveal artifacts. This surface-based representation supports advanced analyses and visualization techniques, potentially enhancing mission design by providing intuitive, analyzable surfaces for high-dimensional solution spaces.

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.

Figures (3)

• Figure 1: The CR3BP is shown in the rotating frame. The Earth is $m_1$, represented by the blue dot, and the Moon is $m_2$, represented by the purple dot. $m_1$ and $m_2$ are located on the x-axis at 0 and 1 respectively. The dark red dot labeled s/c represents the spacecraft, $m_3$, on its approach. The libration points L1, L2, and L3 are represented by red dots. The libration points L4 and L5 are represented by orange dots. The green dashed line shows the orbit of $m_2$ about $m_1$. The yellow dotted lines illustrate the location of L4 and L5 at the third vertex of the equilateral triangles formed by $m_1$, $m_2$, and either libration point.
• Figure 2: In a KNNG with toroidal topology (edges shown in orange), the two largest cycles in the MCB are considered the non-trivial cycles. They capture the periodicity of a toroidal manifold along both the toroidal and poloidal axes. The toroidal cycle (edges shown in green) encircles the empty space at the center of the torus, taking the long way to return to its start vertex. The poloidal cycle (edges shown in blue) encircles the surface of the torus itself, taking the short way to return to its start vertex. The numerically computed point set is shown in black.
• Figure 3: The high-dimensional manifolds are meshed in their native embedding space using the parameterization. The resulting meshes are down-projected to 3D for visualization. Triangle edges are shown in yellow. Triangle color indicates mesh sidedness to aid interpretation of the self-intersection artifacts resulting from the down-projection. The numerically computed point set is shown in black.