In the space $X$ ($X=\mathbb R^2,\mathbb R^3$), there are a family of pairwise disjoint convex closed regions $G_i$ and a shortest trajectory $\cal T$ connecting given initial and finite points and enveloping the regions $G_i$, $\cal T\cap \cup_i \stackrel{\circ} G_i=\varnothing$. Two objects, $t$ and $T$, move under observation along the trajectory $\cal T$ with a constant speed, and the distance $\rho(t,T)$ between the objects along the curve $\cal T$ satisfies the condition $0<\rho(t,T)\le d$ for given $d>0$. We construct a trajectory $\cal T_f$ of the observer's motion and find the observer's speed mode such that the following inequality holds at any time $\tau$ for given $\delta>d$:
$$ \min\big\{\|f_{\tau}-t_{\tau}\|,\|f_{\tau}-T_{\tau}\|\big\}=\delta.$$
Keywords: moving object, observer, trajectory, speed mode
Received August 31, 2022
Revised September 19, 2022
Accepted September 26, 2022
Funding Agency: This study is a part of the research carried out at the Ural Mathematical Center and supported by the Ministry of Science and Higher Education of the Russian Federation (agreement no. 075-02-2022-874).
Vitalii Ivanovich Berdyshev, RAS Academician, Krasovskii Institute of Mathematics and Mechanics of the Ural Branch of the Russian Academy of Sciences, Yekaterinburg, 620108 Russia, e-mail: bvi@imm.uran.ru
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Cite this article as: V.I. Berdyshev. An observer and a pair of objects enveloping a set of convex regions. Trudy Instituta Matematiki i Mekhaniki UrO RAN, 2022, vol. 28, no. 4, pp. 64–70
