Garrick Lau, University of Colorado Boulder; Marcus Holzinger, University of Colorado Boulder; Jill Bruer, Air Force Research Laboratory
Keywords: visualization, differential geometry, maps
Abstract:
Astrography extends geography into space regimes to communicate best paths in space through creation of maps. Because of the dynamics of space regimes, the surfaces being examined are of higher dimensions and its geometry is less well-understood than that of the Earth’s surface. The geographic surfaces on the Earth follow spherical geometry, and the visualization of these surfaces can be a globe or a projection of the sphere onto a planar, table-top map. In the latter, the projection distorts geometric quantities, but different morphisms can allow for preservation of certain quantities, such as the angle between two paths along the Earth being preserved in conformal maps like the Mercator projection. In these projections, however, other quantities such as great circle distances are distorted. Except in rare exceptions such as great circles from a singular central point on stereographic projections, the geodesic shape through many projections will tend to be distorted as a result of needing to use different coordinates on the map to convey the information contained within the map [1]. The same issue will appear in projections of astrographic manifolds; when visualizing the surfaces that are collections of trajectories for a specific objective of a space mission, the astrographic map maker will have to determine the best coordinates for communication of best solutions and the shape of the surface.
Crucial to visualization and the selection of the coordinates is the nature of the geodesic. On the surface of the spherical Earth, when measuring shortest physical distance, the geodesic is the great circle; on a flat plane, the geodesic is the Euclidean measurement, which is a straight line. On an astrographic surface, the physical distance is complicated by the fact that the state space and environment space of the surface includes both position and velocity, and often time as well. More importantly, physical distances are not the most important quantities for determining best paths, unlike the great circle on the Earth and a line on a plane, because of the dynamic nature of the astrographic surface. Instead, a separate distance metric should be quantified from the objectives of the space mission, and this will indicate what is a “shortest” distance. Proximity between points on the astrographic surface will be determined by magnitude of trade-off in this distance metric value function.
After the distance metric is determined per goal of space mission, the best coordinates can be determined to both handle the projection from the high dimensions to a visual map and communicate the redefinition of distances as per the distance metric. Both properties, coordinates and distance metric, are contained within a metric tensor [2]. But because of the astrographic dynamics, the metric tensor will again add complexity to the geometry of the Earth, and it will not follow a uniform formula. Instead, the metric tensor will be state-dependent, inheriting a separate formula at each point on the astrographic surface, which often will be estimated through numerical integration methods because of the lack of closed-form expressions of the dynamics. This means that locally, the geometric curvature of the solution surface may be simplified to approach zero, allowing for practical approximations of the surfaces, accurate if the neighborhoods of locality are selected appropriately [3]. Once the metric tensor field is compiled at discretized grids across the surface, the surface can be generated and its shape quantified; this allows for a projection mapping to a visual surface. This research will develop the process of computing distance metrics from space mission objectives and the metric tensor fields for creation of astrographic maps that will assist mission planners and operators.
References
[1] Goldberg, D. M., & Gott III, J. R. (2007). Flexion and skewness in map projections of the earth. arXiv:Astro-ph/0608501.
[2] Qiu, Jiaming, and Xiongtao Dai. 2023. “Estimating Riemannian Metric with Noise-Contaminated Intrinsic Distance.” In Thirty-seventh Conference on Neural Information Processing Systems.
[3] Ginoux, Jean-Marc, and Bruno Rossetto. 2006. “DIFFERENTIAL GEOMETRY AND MECHANICS: APPLICATIONS TO CHAOTIC DYNAMICAL SYSTEMS.” International Journal of Bifurcation and Chaos 16 (04): 887–910.
Approved for public release; distribution is unlimited. Public Affairs release approval AFRL-2024-0838. The views expressed are those of the authors and do not reflect the official guidance or position of the United States Government, the Department of Defense or of the United States Air Force.
Date of Conference: September 17-20, 2024
Track: Astrodynamics