Tracking a vehicle moving with uniform velocity?

Question

Suppose there are three cell towers at three positions $P_1$, $P_2$ and $P_3$. A vehicle is moving at uniform speed along a straight line. Three towers are pinging the vehicle at certain time and obtaining its distance (e.g., at time $t_1$, $P_1$ gets $r_1$, at $t_2$, $P_2$ gets $r_2$ and so on). There are at least nine such pings, three for each tower, and possibly more. From this, can one derive the trajectory of the vehicle and its velocity?

Answer 1

Nothing is said, so I assume a 3D space.

We have positions $P_i$ of the towers and of a vehicle at $x$. We measure distances $r_i(t)$, which correspond to the radii of spheres around the towers.

The intersection of two spheres is a circle, three circles can intersect in a point.

(Large Version)

We need to determine the trajectory of the vehicle, which is a line. For this we need to have two points $x_1 = x(t_1)$ and $x_2 = x(t_2)$.

To form a line, the given uniform speed of the vehicle has to be $v \ne 0$.

Two localizations together with the times should be enough to determine the velocity. So at first look, two rounds of one ping each should be sufficient.

The question is if there are locations, where one round of ping is not enough to localize the vehicle.