Methods are given for tracking (illuminating) by an observer $f$ a stream of objects $\mathbb T$ moving along a trajectory $\cal T$ that is equipped for temporary stopping of objects $\mathbb T$ at checkpoints $t\in {\cal T}$ and is composed of straight-line segments $\Delta$ and arcs $\Lambda$ circumventing complex regions of the Earth's surface by means of known geometric constructions (ellipsoid, cylinder, prism, etc.) with known geodesic lines (their discrete analogues). In case of traffic congestion, an additional trajectory ${\cal T}^1$ and methods for the rapid transition of the observer from trajectory $\cal T$ to a section of trajectory ${\cal T}^1$, or to the simultaneous illumination of objects ${\mathbb T}\in {\cal T},\ {\mathbb T}^1\subset {\cal T}^1$, are introduced. The observer can simultaneously illuminate a pair of objects located on opposite sides of it. Upon detecting the observer $f$, a hostile object $\mathbb T$ can direct a mini-object dangerous to $f$ toward it; the observer's task is to avoid an encounter with it.
Keywords: navigation, moving object, trajectory, observer, autonomous agent
Received May 18, 2026
Revised May 23, 2026
Accepted May 29, 2026
Funding Agency: The work was performed as part of research conducted in the Ural Mathematical Center with the financial support of the Ministry of Science and Higher Education of the Russian Federation (Agreement number № 075-02-2026-737).
Vitalii Ivanovich Berdyshev, RAS Academician, Krasovskii Institute of Mathematics and Mechanics of the Ural Branch of the Russian Academy of Sciences, Yekaterinburg, 620077 Russia, e-mail: bvi@imm.uran.ru
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Cite this article as: V.I. Berdyshev. Simultaneous streams of objects circumventing obstacles in $\mathbb{R}^3$ under observation. Trudy Instituta Matematiki i Mekhaniki UrO RAN, 2026, vol. 32, no. 3, pp. 21–29.
