An observer and a pair of objects enveloping a set of convex regions

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 $\mathcal T$ connecting given initial and finite points and enveloping the regions $G_i$, $\mathcal T\cap \cup_i \stackrel{\circ} G_i=\varnothing$. Two objects, $t$ and $T$, move under observation along the trajectory $\mathcal T$ with a constant speed, and the distance $\rho(t,T)$ between the objects along the curve $\mathcal T$ satisfies the condition $0<\rho(t,T)\le d$ for given $d>0$. We construct a trajectory $\mathcal T_f$ of the observer's motion and find the observer's speed mode such that the following inequality holds at any time $\tau$ for given $\delta>d$:
$$ \min\big\{\|f_{\tau}-t_{\tau}\|,\|f_{\tau}-T_{\tau}\|\big\}=\delta. $$