Trajectory of an Observer Tracking the Motion of an Object around a Convex Set in ${{\mathbb{R}}^{{\mathbf{3}}}}$

Abstract—An object t moving in ${{\mathbb{R}}^{3}}$ goes around a solid convex set along the shortest path $\mathcal{T}$ under observation. The task of an observer f (moving at the same speed as the object) is to find a trajectory closest to $\mathcal{T}$ that satisfies the condition $\delta \leqslant {\text{||}}f - t{\text{||}} \leqslant K\delta $ for a given $\delta > 0$. This condition makes it possible to track the object along the entire trajectory $\mathcal{T}$. A method is proposed for constructing an observer trajectory that ensures that the indicated inequality holds with a constant $K$ arbitrarily close to unity and the object can be observed on its trajectory $\mathcal{T}$, except for an arbitrarily small segment of $\mathcal{T}$.