At radio frequencies, angular measurements, with the precision required for navigation, are not feasible, except if one employs a very large interferometer. But interferometric methods are too cumbersome for navigation. Hence, we have to devise methods employing the measurement of distances, alone.
On a plane surface, the locus of a given distance from a given point, is a circle with the given radius and center. Then a second such circle will intersect the first in two points. A third circle will remove this ambiguity. And a fourth circle is desirable as an error-check.
If either depth or elevation is desired, that is if the problem is in three dimensions, one additional now sphere is required.
However, the measurement of distance requires either a reflector (or preferably, a transponder), so one can measure the round-trip distance. The requirement for an active role on the part of the reference stations places them in the undesirable situation of being subject to saturation (exhaustion of their resources).
If clocks were perfect, we could transmit a pulse at an agreed-upon time from the reference stations. Then the time of the reception of the pulse from each station would indicate the distance to each station.
However, your clock drifts. How do you measure distance and the time-of-day, both?

Well, time-of-day is easy, once you know your position, hence the propagation delay from each reference station. Good.
But, how do you measure distance, without knowing your own time?
Well, all you can measure is the difference in the arrival times of the pulses from the reference stations. Then, the locus becomes a hyperbola (in two dimensions) or a hyperboloid of revolution of two sheets (in three dimensions). Observe that we require one additional reference station. Only one of the branches of the hyperbola -- or sheets of the hyperboloid -- Is pertinent.
Thus, in three-dimensional space, five reference stations are required, plus additional ones for error-checking, redundancy, and to compensate (at least in part) for an unfavorable pattern of deployment of the reference stations and for the possibly unfavorable viewing angle of the reference stations.
This is the problem that the GPS (= Global Positioning System) navigational device has to solve. The Sonobuoy has the same problem when it is operating in passive mode. In active mode, it needs one less Sonobuoy and the Sonobuoys will employ spheres.
Modern clocks are extremely stable; thus once the local clock is synchronized with the clocks on the GPS satellites, the GPS receiver may switch to employing spheres. The slight drift of the local clock can be corrected from the miss-alignment of the intersection of the redundant spheres.
In summary, the minimal requirements are:


(Time is known)


(Time is unknown)

Two-dimensional Two reference stations

employing circles

Three reference stations

employing hyperbolas

Three-dimensional Three reference stations

employing spheres

Four reference stations

employing hyperboloids of revolution of two sheets

But, as discussed previously, one more reference station is required to select from among the several intersection points. Additional (redundant) reference stations will improve the precision of the determination of the intersection point.

Copywrite (c) 1997 R. I. 'Scibor-Marchochi last modified Sunday 13-th of July 1997.