1. An autonomous object \(t\) and an observer f that is hostile toward t move in space \({{\mathbb{R}}^{3}}\) containing a closed convex solid set \(G\) obstructing motion and visibility. The object follows from an initial point \({{t}_{*}}\) to a final point \(t{\text{*}}({{t}_{*}},t{\text{*}} \notin G)\) and goes around \(G\) along the shortest path \(\mathcal{T} = {{\mathcal{T}}_{t}}\). It is assumed that the object and the observer have identical velocities \({{V}_{t}},~{{V}_{f}}\) and, at each time \(\tau \), the distance between the positions of the mutually visible \(t = {{t}_{\tau }}\) and \(f = {{f}_{\tau }}\) satisfies the inequalities

$$0 < \delta \leqslant {\text{||}}{{t}_{\tau }} - {{f}_{\tau }}{\text{||}} \leqslant K \cdot \delta $$
(1)

for given \(\delta \) and \(K \geqslant 1\). The left inequality “ensures” the mutual safety of the object and the observer, while the right inequality is used to improve the quality of the observation.

The task of the observer is to design a trajectory \({{\mathcal{T}}_{f}}\) such that inequality (1) is satisfied with as small a constant K as possible and the observer \({{f}_{\tau }} \in {{\mathcal{T}}_{f}}\) is able to observe the moving object \({{t}_{\tau }}\) on as long a segment of \({{\mathcal{T}}_{t}}\) as possible.

In the absence of constraints on the observer velocity magnitude \(\left| {{{V}_{f}}} \right|\), the problem is easy to solve. Specifically, moving along the trajectory \({{\mathcal{T}}_{t}} = \left\{ {{{f}_{\tau }}} \right\}\), where \({{f}_{\tau }} = {{t}_{\tau }} - \delta \frac{{{{V}_{{{{t}_{\tau }}}}}}}{{\left| {{{V}_{{{{t}_{\tau }}}}}} \right|}}\), the observer \({{f}_{\tau }}\) tracks the motion of the object \({{t}_{\tau }}\) along the entire trajectory \({{\mathcal{T}}_{t}}\); moreover, \({\text{||}}{{t}_{\tau }} - {{f}_{\tau }}{\text{||}} = \delta \) and the observer velocity \({{V}_{{{{f}_{\tau }}}}}\) depends on \({{V}_{{{{t}_{\tau }}}}}\) and the curvature of \({{\mathcal{T}}_{t}}\). For the observer, it is inexpedient to follow the object along \({{\mathcal{T}}_{t}}\) on its strongly convex segments and near its corner points because of the fear of losing sight of the object \(t\). Since \({{\mathcal{T}}_{t}}\) is the shortest path, whereas \({{\mathcal{T}}_{t}}\) under the conditions \(\left| {{{V}_{t}}} \right| = \left| {{{V}_{f}}} \right|\) and (1) is not, there exists a segment of \({{\mathcal{T}}_{t}}\) on which the object moves unseen over the time \({{t}_{\tau }}\).

In this paper, we propose a method for constructing a trajectory \({{\mathcal{T}}_{t}}\) ensuring that inequality (1) holds with a constant \(K\) arbitrarily close to unity and the unobseved segment of \({{\mathcal{T}}_{t}}\) has an arbitraruly short length.

2. Since \({{t}_{*}}\) and t* are not contained in the set \(G\), the initial and final segments of the trajectory \(\mathcal{T}\) are straight-line segments. Denote them by \([{{t}_{*}},\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{t} ]\) and \(\left[ {t{\text{*}},\bar {t}} \right]\). Additinally, we use the notation

$$\begin{gathered} {{l}_{*}} = \left\{ {{{t}_{*}} + \lambda \left( {\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{t} - {{t}_{*}}} \right){\text{: }}\lambda \geqslant 0} \right\}, \\ l{\text{*}} = \left\{ {t{\text{*}} + \lambda \left( {\bar {t} - t{\text{*}}} \right){\text{: }}\lambda \geqslant 0} \right\}; \\ \end{gathered} $$

\({{P}_{{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{t} ~}}},~{{P}_{{\bar {t}}}}\) are the supporting planes of the set \(G\) at the points \(\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{t} \) and \(\bar {t}\), respectively. Note that \({{l}_{*}} \subset {{P}_{{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{t} ~}}}\) and \(~l{\text{*}} \subset {{P}_{{\bar {t}}}}\).

3. Suppose that the planes \({{P}_{{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{t} ~}}},~{{P}_{{\bar {t}}}}\) are parallel or intersect, \(l = {{P}_{{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{t} ~}}} \cap ~\,\,{{P}_{{\bar {t}}}}\), and

$$\rho \left( {{{t}_{*}},l} \right) > \rho \left( {\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{t} ,l} \right),\quad \rho \left( {t{\text{*}},l} \right) > \rho \left( {\bar {t},l} \right).$$
(2)

Moving along the trajectory

$${{\mathcal{T}}_{f}} = \mathcal{T} + b,\quad {\text{where}}\quad b = \delta \frac{{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{t} - {{t}_{*}}}}{{{\text{||}}\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{t} - {{t}_{*}}{\text{||}}}},$$
(3)

the observer \({{f}_{\tau }}\) is able to see the object tτ = \(({{f}_{\tau }} - b)\,\, \in \,\,\mathcal{T}\) without changing the direction of observation \(b\).

4. Consider the case when the reverses of inequalities (2) hold (see Fig. 1). In what follows, \(\widehat {t,t'}\) is the arc of \(\mathcal{T}\) lying between the points \(t,~t'\) and \(\left| {\widehat {t,t'}} \right|\) is the length of this arc. We introduce a sequence of points \({{t}_{i}} \in \mathcal{T}\) and define the corresponding sequence of points \({{f}_{i}}\). Later, the piecewise linear arc with nodes at \({{f}_{i}}\) will be included in the trajectory \({{\mathcal{T}}_{f}}\).

Fig. 1.
figure 1

Trajectories \(\mathcal{T}\), \({{\mathcal{T}}_{f}}\) (thick curves) of the object and the observer; and the set G and the supporting planes \({{P}_{{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{t} }}}\), \({{P}_{{\bar {t}}}}\) (thin curves).

Full size image

Since \(\mathcal{T}\) is the shortest path, there exists a pair of tangent vectors at each point \(t\) of \(\mathcal{T}\) (see, e.g., [1, 2]). Let \({{L}_{i}}\) denote the tangent vector to \({{V}_{{{{t}_{i}}}}}\) at the point \({{t}_{i}}\), which is the velocity of the object t. The arc \(\widehat {{{t}_{i}},{{t}_{{i + 1}}}}\) of the trajectory \(\mathcal{T}\) is denoted by Δi.

The point \({{t}_{1}} \in \mathcal{T}\) is such that the tangent vector to \(\mathcal{T}\) at this point is orthogonal to the ray \({{l}_{*}}\). Let \({{f}_{1}} = {{t}_{1}} + {{b}_{1}}\), where \({{b}_{1}} = b\) (see (3)). As an initial segment of the trajectory \({{\mathcal{T}}_{f}}\), we use the arc \((\widehat {\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{t} ,{{t}_{1}}}) + {{b}_{1}}\). To determine \({{t}_{2}} \in \widehat {{{t}_{1}},\bar {t}}\), we find a point \(t_{2}^{'} \in \widehat {{{t}_{1}},\bar {t}} \subset \mathcal{T}\) such that the straight line \({{Q}_{1}}\) containing \(t_{2}^{'}\) and parallel to the vector \({{b}_{1}}\) does not intersect \(G^\circ \), where \(G^\circ \) is the interior of the set G. The point t2 must lie on the arc \(\widehat {{{t}_{1}},t_{2}^{'}}\) at a small distance from \({{t}_{1}}\). We construct the arc \(\widehat {{{f}_{1}},f_{2}^{'}} = {{\Delta }_{1}} + {{b}_{1}}\) and, on the ray \({{L}_{1}} + {{f}_{1}}\), mark the point f2 for which \({\text{||}}{{f}_{1}} - {{f}_{2}}{\text{||}} = \left| {{{\Delta }_{1}}} \right|\). A continuous one-to-one mapping of the segment \([{{f}_{1}},{{f}_{2}}]\) to the arc \(\widehat {{{t}_{1}},{{t}_{2}}}\) supplies the observer \({{f}_{\tau }} \in [{{f}_{1}},{{f}_{2}}]\) with a way of tracking the moving object \({{t}_{\tau }}\). This completes the first step. It is easy to see that an increase in the distance from \({{t}_{1}}\) to \({{t}_{2}}\) leads to an an increase in \({\text{||}}{{f}_{2}} - {{t}_{2}}{\text{||}}\) and in the constant K in inequality (1). At the second step, by analogy with the first one, we find a point \(t_{3}^{'} \in \mathcal{T}\) such that the straight line \({{Q}_{2}}\) containing the point \(t_{3}^{'}\) and parallel to the vector \({{b}_{2}} = {{f}_{2}} - {{t}_{2}}\) does not intersect \(G^\circ \). The point \({{t}_{3}}\) is taken on the arc \(\widehat {{{t}_{2}},t_{3}^{'}}\) at a small distance from \({{t}_{2}}\), etc. Increasing the number of steps, we construct sequences \(\left\{ {{{t}_{i}}} \right\} \in \mathcal{T}\), \({{t}_{i}} \to \bar {t}\), and \(\left\{ {{{f}_{i}}} \right\}\), \(\rho \left( {{{f}_{i}},{{P}_{{\bar {t}}}}} \right) \to 0\) \(\left( {i \to \infty } \right)\). The trajectory of the observer f has the form

$${{\mathcal{T}}_{f}} = (\widehat {\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{t} ,{{t}_{1}}} + {{b}_{1}}){{ \cup }_{i}}\,\,\left[ {{{f}_{i}},{{f}_{{i + 1}}}} \right].$$
(4)

The constructed sequences \({{t}_{i}},~{{f}_{i}}~\,\left( {i = 1,2, \ldots } \right)\) satisfy the relations

$${\text{||}}{{t}_{{i + 1}}} - {{f}_{{i + 1}}}{\text{||}} - \,{\text{||}}{{t}_{i}} - {{f}_{i}}{\text{||}} = {\text{||}}{{t}_{{i + 1}}} - {{f}_{{i + 1}}}{\text{||}} - \,{\text{||}}{{t}_{{i + 1}}} - f_{{i + 1}}^{'}{\text{||}}$$
$$ \leqslant {\text{||}}\,f_{{i + 1}}^{'} - {{f}_{{i + 1}}}{\text{||}} = o({\text{||}}\,f_{{i + 1}}^{'} - {{f}_{i}}{\text{||}}) = o\left( {{\text{||}}{{t}_{i}} - {{t}_{{i + 1}}}{\text{||}}} \right),$$
(5)
$${\text{||}}{{t}_{{i + 1}}} - {{f}_{{i + 1}}}{\text{||}} \leqslant {\text{||}}{{t}_{i}} - {{f}_{i}}{\text{||}} + \,{\text{||}}\,f_{{i + 1}}^{'} - {{f}_{{i + 1}}}{\text{||}}$$
$$\begin{gathered} \leqslant {\text{||}}{{t}_{{i - 1}}} - {{f}_{{i - 1}}}{\text{||}} + \,{\text{||}}\,f_{i}^{'} - {{f}_{i}}{\text{||}} \\ + \,\,{\text{||}}\,f_{{i + 1}}^{'} - {{f}_{{i + 1}}}{\text{||}} \leqslant ... \leqslant \delta + \mathop {\sum \,}\limits_2^{i + 1} {\text{||}}\,f_{k}^{'} - {{f}_{k}}{\text{||,}} \\ \end{gathered} $$
(6)

moreover, \({\text{||}}{{t}_{i}} - {{f}_{i}}{\text{||}}\) is an increasing sequence.

Theorem 1. Let \(\left\{ {{{t}_{i}}} \right\}_{1}^{\infty } \subset \widehat {{{t}_{1}},\bar {t}}~\) be the sequence of points generated according to the rule described above, and \(\widehat {{{t}_{{i + 1}}},\bar {t}} \subset \widehat {{{t}_{i}},\bar {t}}\) \((i = 1,2 \ldots ).\) Moving along the trajectory \({{\mathcal{T}}_{f}}\) (4), the observer f is able to track the motion of the object \(t = t\left( f \right)\) on \(\mathcal{T}\), where \(t\left( f \right) = f - {{b}_{1}}\) for \(f\, \in \,(\widehat {\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{t} ,{{t}_{1}}})\) + b1 and \(t({{f}^{\lambda }}) \in {{\Delta }_{i}}\), \({\text{|}}\widehat {{{t}_{i}},t({{f}^{\lambda }})}{\text{|}} = \lambda \left| {{{\Delta }_{i}}} \right|\) for \({{f}^{\lambda }}\, = \,(1\, - \,\lambda ){{f}_{i}}\, + \,\lambda {{f}_{{i + 1}}}\) \(\left( {0 \leqslant \lambda \leqslant 1} \right)\).

The number of segments in (4) can be limited by projecting the point fj for a sufficiently large i onto the plane \({{P}_{{\bar {t}}}}\). Denote this projection by \(\bar {f}\). While moving along the segment \(\left[ {{{t}_{j}},\bar {t}} \right],\) the observer does not see the object following the arc \(\widehat {{{f}_{j}},\bar {f}}\), but, while moving on the plane \({{P}_{{\bar {t}}}}\) from the position \({{f}_{\tau }} = \bar {f}\), the observer tracks the motion of the object over the segment \(\left[ {\bar {t},t{\text{*}}} \right]\).

By using inequalities (5) and (6), it is easy to prove the following result.

Theorem 2. Suppose that, for any index \(n = 1,2, \ldots \) , the rule described above generates an ordered grid of nodes \(\left\{ {t_{i}^{n}} \right\}_{{i = 1}}^{{k\left( n \right)}} \subset \widehat {{{t}_{1}},\bar {t}}~\) such that \(t_{{k\left( n \right)}}^{n} \to \bar {t}\) as \(k\left( n \right) \to \infty \) and

$${\text{|}}\widehat {t_{i}^{n},t_{{i + 1}}^{n}}{\text{|}} \leqslant \frac{{\left| {\widehat {{{t}_{1}},\bar {t}}} \right|}}{n}.$$

Then, for the sequence of trajectories

$${{\mathcal{T}}_{{{{f}^{n}}}}} = (\widehat {\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{t} ,{{t}_{1}}} + {{b}_{1}})\bigcup\limits_{i = 1}^{k\left( n \right) - 1} {[f_{i}^{n},f_{{i + 1}}^{n}]\quad \left( {n = 1,2, \ldots } \right),} $$

where \(f_{i}^{n} = t_{i}^{n} + {{b}_{i}}\), it is true that

$$\mathop {\max }\limits_i {\text{||}}t_{i}^{n} - f_{i}^{n}{\text{||}} \to \delta \quad as\quad n \to \infty .$$