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

2025-04-04

AI Summary

Addressing the challenge of modeling and visualizing high-dimensional toroidal manifolds—arising from quasi-periodic three-body orbits in the Circular Restricted Three-Body Problem (CR3BP)—this paper proposes an embedding-independent mesh generation method driven by discrete 1-forms. The method achieves, for the first time, topologically faithful triangulation of such toroidal manifolds and introduces a triangle orientation assignment algorithm tailored for dimensionality-reduced visualization, markedly enhancing projection interpretability. Grounded in discrete differential geometry and point-cloud topological parameterization theory, it constructs the first continuous, renderable 3D surface models—elevating conventional representations (discrete points or curves) to geometrically complete surface descriptions. The resulting models enable visual analysis and interactive trajectory design for space missions, establishing a novel paradigm for interpretable, high-dimensional dynamical structure modeling.

Technology Category

• Machine Learning: Learning with Manifolds
• Computer Vision: 3D Computer Vision
• Knowledge Representation and Reasoning: Geometric, Spatial, and Temporal Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• User Modeling, Personalization and Recommendation: Explainable and interpretable methods for personalization
• Web Mining and Content Analysis: Web data visualization

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.

Problem

• Mesh high-dimensional toroidal manifolds from quasi-periodic three-body dynamics
• Enable accurate modeling and visualization of high-dimensional manifolds
• Enhance interpretation of meshes via embedding-agnostic triangle-sidedness algorithm

Innovation

• Embedding-agnostic parameterization for toroidal manifolds
• Discrete one-form-based meshing for high-dimensional point clouds
• Triangle-sidedness algorithm enhances 3D visualization