Editing Global Positioning System (section)

=== Geometric interpretation ===
The GPS equations can be solved by numerical and analytical methods. Geometrical interpretations can enhance the understanding of these solution methods.

==== Spheres ====
[[File:2D Trilat Scenario 2019-0116.jpg|thumb|2-D Cartesian true-range multilateration (trilateration) scenario]]

The measured ranges, called pseudoranges, contain clock errors. In a simplified idealization in which the ranges are synchronized, these true ranges represent the radii of spheres, each centered on one of the transmitting satellites. The solution for the position of the receiver is then at the intersection of the surfaces of these spheres; see [[trilateration]] (more generally, true-range multilateration). Signals from at minimum three satellites are required, and their three spheres would typically intersect at two points.<ref>{{cite web|url=http://www.math.nus.edu.sg/aslaksen/gem-projects/hm/0203-1-10-instruments/modern.htm|title=Modern navigation|work=math.nus.edu.sg|access-date=December 4, 2018|archive-url=https://web.archive.org/web/20171226024421/http://www.math.nus.edu.sg/aslaksen/gem-projects/hm/0203-1-10-instruments/modern.htm|archive-date=December 26, 2017}}</ref> One of the points is the location of the receiver, and the other moves rapidly in successive measurements and would not usually be on Earth's surface.

In practice, there are many sources of inaccuracy besides clock bias, including random errors as well as the potential for precision loss from subtracting numbers close to each other if the centers of the spheres are relatively close together. This means that the position calculated from three satellites alone is unlikely to be accurate enough. Data from more satellites can help because of the tendency for random errors to cancel out and also by giving a larger spread between the sphere centers. But at the same time, more spheres will not generally intersect at one point. Therefore, a near intersection gets computed, typically via least squares. The more signals available, the better the approximation is likely to be.

==== Hyperboloids ====
[[File:Hyperbolic Navigation.svg|thumb|219x219px|Three satellites (labeled as "stations" A, B, C) have known locations. The true times it takes for a radio signal to travel from each satellite to the receiver are unknown, but the true time differences are known. Then, each time difference locates the receiver on a branch of a hyperbola focused on the satellites. The receiver is then located at one of the two intersections.]]

If the pseudorange between the receiver and satellite ''i'' and the pseudorange between the receiver and satellite ''j'' are subtracted, {{nowrap|1=''p<sub>i</sub>'' − ''p<sub>j</sub>''}}, the common receiver clock bias (''b'') cancels out, resulting in a difference of distances {{nowrap|1=''d<sub>i</sub>'' − ''d<sub>j</sub>''}}. The locus of points having a constant difference in distance to two points (here, two satellites) is a [[hyperbola]] on a plane and a [[hyperboloid of revolution]] (more specifically, a [[two-sheeted hyperboloid]]) in 3D space (see [[Multilateration]]). Thus, from four pseudorange measurements, the receiver can be placed at the intersection of the surfaces of three hyperboloids each with [[Focus (geometry)|foci]] at a pair of satellites. With additional satellites, the multiple intersections are not necessarily unique, and a best-fitting solution is sought instead.<ref name="Abel1" /><ref name="Fang" /><ref>{{cite book |last1=Strang |first1=Gilbert |url=https://books.google.com/books?id=MjNwWUY8jx4C&pg=PA449 |title=Linear Algebra, Geodesy, and GPS |last2=Borre |first2=Kai |publisher=SIAM |year=1997 |isbn=978-0-9614088-6-2 |pages=448–449 |access-date=May 22, 2018 |archive-url=https://web.archive.org/web/20211010021202/https://books.google.com/books?id=MjNwWUY8jx4C&pg=PA449 |archive-date=October 10, 2021 |url-status=live}}</ref><ref>{{cite book |author=Holme |first=Audun |url=https://books.google.com/books?id=zXwQGo8jyHUC&pg=PA338 |title=Geometry: Our Cultural Heritage |publisher=Springer Science & Business Media |year=2010 |isbn=978-3-642-14441-7 |page=338 |access-date=May 22, 2018 |archive-url=https://web.archive.org/web/20211010021203/https://books.google.com/books?id=zXwQGo8jyHUC&pg=PA338 |archive-date=October 10, 2021 |url-status=live}}</ref><ref name="HWLW">{{cite book |last1=Hofmann-Wellenhof |first1=B. |url=https://books.google.com/books?id=losWr9UDRasC&pg=PA36 |title=Navigation |last2=Legat |first2=K. |last3=Wieser |first3=M. |publisher=Springer Science & Business Media |year=2003 |isbn=978-3-211-00828-7 |page=36 |access-date=May 22, 2018 |archive-url=https://web.archive.org/web/20211010021203/https://books.google.com/books?id=losWr9UDRasC&pg=PA36 |archive-date=October 10, 2021 |url-status=live}}</ref><ref name="Groves2013">{{cite book |last=Groves |first=P. D. |url=https://books.google.com/books?id=t94fAgAAQBAJ |title=Principles of GNSS, Inertial, and Multisensor Integrated Navigation Systems, Second Edition |publisher=Artech House |year=2013 |isbn=978-1-60807-005-3 |series=GNSS/GPS |page= |access-date=February 19, 2021 |archive-url=https://web.archive.org/web/20210315202930/https://books.google.com/books?id=t94fAgAAQBAJ |archive-date=March 15, 2021 |url-status=live}}</ref>

==== Inscribed sphere ====
[[File:Descartes Circles.svg|thumb|A smaller circle ({{color|red|'''red'''}}) inscribed and tangent to other circles ({{color|black|'''black'''}}), that need not necessarily be mutually tangent]]

The receiver position can be interpreted as the center of an [[inscribed sphere]] (insphere) of radius ''bc'', given by the receiver clock bias ''b'' (scaled by the speed of light ''c''). The insphere location is such that it touches other spheres. The [[Circumscribed sphere|circumscribing spheres]] are centered at the GPS satellites, whose radii equal the measured pseudoranges ''p''<sub>i</sub>. This configuration is distinct from the one described above, in which the spheres' radii were the unbiased or geometric ranges ''d''<sub>i</sub>.<ref name=HWLW />{{rp|36–37}}<ref name="Hoshen 1996">{{cite journal |author=Hoshen |first=J. |year=1996 |title=The GPS Equations and the Problem of Apollonius |journal=IEEE Transactions on Aerospace and Electronic Systems |volume=32 |issue=3 |pages=1116–1124 |bibcode=1996ITAES..32.1116H |doi=10.1109/7.532270 |s2cid=30190437}}</ref>

==== Hypercones ====
The clock in the receiver is usually not of the same quality as the ones in the satellites and will not be accurately synchronized to them. This produces [[pseudorange]]s with large differences compared to the true distances to the satellites. Therefore, in practice, the time difference between the receiver clock and the satellite time is defined as an unknown clock bias ''b''. The equations are then solved simultaneously for the receiver position and the clock bias. The solution space [''x, y, z, b''] can be seen as a four-dimensional [[spacetime]], and signals from at minimum four satellites are needed. In that case each of the equations describes a [[hypercone]] (or spherical cone),<ref>{{cite journal|title=GPS Solutions: Closed Forms, Critical and Special Configurations of P4P | doi=10.1007/PL00012897 | volume=5|issue=3 |journal=GPS Solutions|pages=29–41 | last1 = Grafarend | first1 = Erik W.|year=2002 | bibcode=2002GPSS....5...29G | s2cid=121336108 }}</ref> with the cusp located at the satellite, and the base a sphere around the satellite. The receiver is at the intersection of four or more of such hypercones.