Abstract
The paper considers the solution of the reduction of the geometry of motion from a three-dimensional surface to the plane. Parallel lines on the Lobachevsky plane immersed in three-dimensional Euclidean space are considered as initial routes. The geometrical place of route points in three-dimensional space is defined as the geodesic line of the pseudosphere. The result of the reduction are curves on the Euclidean plane with Riemannian metric. The geometrical reduction of motion implies the following goal of the produced transformations - the translation of spatial trajectories into numerical manifolds of two-dimensional dimension, formalized as matrices, affinors of linear transformations, and valence tensors two. The resulting manifolds can be taken as input data for software processing and obtaining results of trajectory state analysis. For example, in the form of a metric estimate of the geometric location distribution of points based on cluster analysis. The obtained results may find application in mechanics applications investigating motion along surfaces of short waves (solitons) as well as on surfaces obtained by immersing parts of the Lobachevsky plane into the three-dimensional Euclidean space. In addition, it should be taken into account that this immersion is related to the behavior of evolution equations.
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Gushin, A., Chertykovtseva, N., Palevskaya, S., Pavlova, O., Gulenko, O. (2023). Mapping Motion Paths from Non-zero Curvature Surfaces. In: Beskopylny, A., Shamtsyan, M., Artiukh, V. (eds) XV International Scientific Conference “INTERAGROMASH 2022”. INTERAGROMASH 2022. Lecture Notes in Networks and Systems, vol 575. Springer, Cham. https://doi.org/10.1007/978-3-031-21219-2_25
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DOI: https://doi.org/10.1007/978-3-031-21219-2_25
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