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 errorcheck.
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 roundtrip
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
agreedupon 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
timeofday, both?
Well, timeofday 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 threedimensional space, five reference stations are
required, plus additional ones for errorchecking, 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 missalignment of the
intersection of the redundant spheres.
In summary, the minimal requirements are:
Active (Time is known) 
Passive (Time is unknown) 

Twodimensional  Two reference stations employing circles 
Three reference stations employing hyperbolas 
Threedimensional  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. 'SciborMarchochi last modified Sunday 13th of July 1997.