Editing True-range multilateration (section)

=== Two spherical dimensions, two or more measured spherical ranges ===
[[Image:Sun Moon (annotated).gif|right|thumb|300px|Fig. 3 Example of celestial navigation altitude intercept problem (lines of position are distorted by the map projection)]]
This is a classic celestial (or astronomical) navigation problem, termed the ''altitude intercept'' problem (Fig. 3). It's the spherical geometry equivalent of the trilateration method of surveying (although the distances involved are generally much larger). A solution at sea (not necessarily involving the Sun and Moon) was made possible by the [[marine chronometer]] (introduced in 1761) and the discovery of the 'line of position' (LOP) in 1837. The solution method now most taught at universities (e.g., U.S. Naval Academy) employs [[spherical trigonometry]] to solve an oblique spherical triangle based on [[sextant]] measurements of the 'altitude' of two heavenly bodies.<ref name="Todhunter">[https://www.gutenberg.org/files/19770/19770-pdf.pdf ''Spherical Trigonometry''], Isaac Todhunter, MacMillan; 5th edition, 1886.</ref><ref name="Casey">''A treatise on spherical trigonometry, and its application to geodesy and astronomy, with numerous examples'', John Casey, Dublin, Hodges, Figgis & Co., 1889.</ref> This problem can also be addressed using vector analysis.<ref name="Veness">[https://www.movable-type.co.uk/scripts/latlong-vectors.html "Vector-based geodesy"], Chris Veness. 2016.</ref> Historically, graphical techniques – e.g., the [[intercept method]] – were employed. These can accommodate more than two measured 'altitudes'. Owing to the difficulty of making measurements at sea, 3 to 5 'altitudes' are often recommended.

As the earth is better modeled as an ellipsoid of revolution than a sphere, iterative techniques may be used in modern implementations.<ref name="Kaplan">"STELLA (System To Estimate Latitude and Longitude Astronomically)", George Kaplan, John Bangert, Nancy Oliversen; U.S. Naval Observatory, 1999.</ref> In high-altitude aircraft and missiles, a celestial navigation subsystem is often integrated with an inertial navigation subsystem to perform automated navigation—e.g., U.S. Air Force [[Lockheed SR-71 Blackbird|SR-71 Blackbird]] and [[Northrop Grumman B-2 Spirit|B-2 Spirit]].

While intended as a 'spherical' pseudo range multilateration system, Loran-C has also been used as a 'spherical' true-range multilateration system by well-equipped users (e.g., Canadian Hydrographic Service).<ref name="Grant" /> This enabled the coverage area of a Loran-C station triad to be extended significantly (e.g., doubled or tripled) and the minimum number of available transmitters to be reduced from three to two. In modern aviation, slant ranges rather than spherical ranges are more often measured; however, when aircraft altitude is known, slant ranges are readily converted to spherical ranges.<ref name="Geyer" />