Editing True-range multilateration (section)

{{Short description|Using distance measures along a shape's edges to determine position in space}}
{{broader|Multilateration}}
{{more citations needed|date=June 2017}}

'''True-range multilateration''' (also termed '''range-range multilateration''' and '''spherical multilateration''') is a method to determine the location of a movable vehicle or stationary point in space using multiple [[ranging|ranges]] ([[distance]]s) between the vehicle/point and multiple spatially-separated known locations (often termed "stations"). <ref name="Lee">[https://rosap.ntl.bts.gov/view/dot/12134 ''Accuracy limitations of range-range (spherical) multilateration systems''], Harry B. Lee, Massachusetts Institute of Technology, Lincoln Laboratory, Report Number: DOT/TSC-RA-3-8-(1) (Technical note 1973-43), Oct. 11, 1973</ref><ref name="Grant" /> Energy waves may be involved in determining range, but are not required.

True-range multilateration is both a mathematical topic and an applied technique used in several fields. A practical application involving a fixed location occurs in [[surveying]].<ref name="Wirtanen 1969 pp. 81–92">{{cite journal | last=Wirtanen | first=Theodore H. | title=Laser Multilateration | journal=Journal of the Surveying and Mapping Division | publisher=American Society of Civil Engineers (ASCE) | volume=95 | issue=1 | year=1969 | issn=0569-8073 | doi=10.1061/jsueax.0000322 | pages=81–92}}</ref><ref name="Escobal Fliegel Jaffe Muller 2013 p.">{{cite journal | last=Escobal | first=P. R. | last2=Fliegel | first2=H. F. | last3=Jaffe | first3=R. M. | last4=Muller | first4=P. M. | last5=Ong | first5=K. M. | last6=Vonroos | first6=O. H. | title=A 3-D Multilateration: A Precision Geodetic Measurement System | journal=JPL Quart. Tech. Rev. | volume=2 | issue=3 | date=2013-08-07 | url=https://ntrs.nasa.gov/citations/19730002255 | access-date=2022-11-06 | page=}}</ref> Applications involving vehicle location are termed [[navigation]] when on-board persons/equipment are informed of its location, and are termed [[surveillance]] when off-vehicle entities are informed of the vehicle's location.

Two ''[[slant range]]s'' from two known locations can be used to locate a third point in a two-dimensional Cartesian space (plane), which is a frequently applied technique (e.g., in surveying). Similarly, two ''[[spherical range]]s'' can be used to locate a point on a sphere, which is a fundamental concept of the ancient discipline of [[celestial navigation]] — termed the ''altitude intercept'' problem. Moreover, if more than the minimum number of ranges are available, it is good practice to utilize those as well. This article addresses the general issue of position determination using multiple ranges.
 
In [[2D geometric model|two-dimensional geometry]], it is known that if a point lies on two circles, then the circle centers and the two radii provide sufficient information to narrow the possible locations down to two – one of which is the desired solution and the other is an ambiguous solution. Additional information often narrow the possibilities down to a unique location. In three-dimensional geometry, when it is known that a point lies on the surfaces of three spheres, then the centers of the three spheres along with their radii also provide sufficient information to narrow the possible locations down to no more than two (unless the centers lie on a straight line).

True-range multilateration can be contrasted to the more frequently encountered [[pseudo-range multilateration]], which employs range differences to locate a (typically, movable) point. Pseudo range multilateration is almost always implemented by measuring [[Time of arrival|times-of-arrival]] (TOAs) of energy waves. True-range multilateration can also be contrasted to [[triangulation (surveying)|triangulation]], which involves the measurement of [[angle]]s.