File:Hyperbolic Navigation.svg

English: This is the geometry of a basic hyperbolic navigation system. Hyperbolic navigation is a class of navigation systems based on the difference in timing between the reception of two signals, without reference to a common clock.

Here, the three ground stations are Stations A, B, C, whose locations are known. The times it takes for a radio signal to travel from the stations to the receiver are unknown, but the time differences are known. That is, ${\displaystyle t_{A},t_{B},t_{C}}$ are unknown, but ${\displaystyle t_{A}-t_{B}}$ and ${\displaystyle t_{A}-t_{C}}$ are known.

Then, each time difference locates the receiver on a branch of a hyperbola focussed on the ground stations. Then the location of the receiver is at one of the two intersections. Any other form of navigation information can be used to eliminate this ambiguity and determine a fix.

Created in GeoGebra. Permanent link here.

Additional note: We assume that the earth is approximately flat in this diagram. This assumption can be corrected. In general, the surface of equal time difference is a hyperboloid in space, and it intersects with a sphere (the surface of earth) at an ellipse. To see this, note that a hyperboloid

To locate a point uniquely in space, one must use at least least 4 ground stations. The receiver can then be located as one of the unique intersections of the 6 hyperboloids (one hyperboloid for each pair of the ground stations).