Editing Global Positioning System (section)

==== 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>