Interference pattern from two transmitters
Hyperbolic stand lines (black lines) between two transmitters

Under the term hyperbola navigation or hyperbola localization , geometric-physical methods of location determination are summarized in which either

• Distances between discrete points are measured.
For example, it can be used
• that the received field strength E of an electromagnetic wave decreases approximately inversely proportional to the distance d (d ~ 1 / E)
• that the transit time of the signal increases with distance
• or speed measurements can be integrated over time using the Doppler effect .

Each tuple of several measurements provides a geometric location for the position of the observer (i.e. his receiver ) if the locations of the transmitters (or the answering transponders ) are known. The number M required measurements is at least 1 higher than the dimension n of the model ( ). ${\ displaystyle M \ geq n + 1}$

## Geometry of the transmitter and receiver positions

Usually these are transmitters or transponders

The geometric locations of each measured difference in distance (also called base lines on defined areas ) are:

• Hyperbola , if the geometric position of all transmitters and receivers can be given by a plane or approximated by it (i.e. on limited parts of the earth's surface). The focal points of these hyperbolas coincide with the positions of the transmitters.
• Hyperboloid of revolution , if the geometric problem cannot be solved on the plane, but in 3D space. In the two focal points of each hyperboloid there is in turn one of the transmitters or the transponder ( code answering machine).

## Solution on the earth's surface or in space

In the case of a flat problem (e.g. in short and medium-range navigation with HiFix or DECCA ), three transmitters are sufficient - which results in two hyperbolas - and the clear reception of your coded signals. The location of the receiver results from the intersection of the two hyperbola stand lines . The third geometric location is generally the surface of the earth or (in navigation) the sheet (s) of a nautical chart or a suitable aeronautical chart (see ICAO and Decca charts).

The spatial intersection requires four transmitters (i.e. four satellites ) which must be computable with sufficient accuracy. If a location is sufficient for only about ± 1 km, then the most up-to-date (" osculating ") orbital elements are sufficient (five geometric elements, a time and generally two rotation rates due to the earth flattening ). If the accuracy is to be higher, one must take into account up to a few hundred orbital parameters and the small irregularities of the earth's rotation .