Trajectory of an observer tracking the motion of an object around a convex set in $\mathbb{R}^3$

An object $t$ moving in $\mathbb{R}^3$ goes around a solid convex set along the shortest path $\mathscr{T}$ under observation. The task of an observer $f$ (moving at the same speed as the object) is to find a trajectory closest to $\mathscr{T}$ that satisfies the condition $\delta\le\|f-t\|\le K\cdot\delta$ for a given $\delta>0$. This condition makes it possible to track the object along the entire trajectory $\mathscr{T}$. 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 $\mathscr{T}$, except for an arbitrarily small segment of $\mathscr{T}$.