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Vincenty's formulae are two related iterative methods used in geodesy to calculate the distance between two points on the surface of a spheroid, developed by Thaddeus Vincenty (1975a). They are based on the assumption that the figure of the Earth is an oblate spheroid, and hence are more accurate than methods that assume a spherical Earth, such ...
Vincenty (1975) provides solutions for the direct and inverse problems; these are based on a series expansion carried out to third order in the flattening and provide an accuracy of about 0.1 mm for the WGS84 ellipsoid; however the inverse method fails to converge for nearly antipodal points.
Finding the geodesic between two points on the Earth, the so-called inverse geodetic problem, was the focus of many mathematicians and geodesists over the course of the 18th and 19th centuries with major contributions by Clairaut, [5] Legendre, [6] Bessel, [7] and Helmert English translation of Astron. Nachr. 4, 241–254 (1825). Errata. [8]
Thaddeus Vincenty (born Tadeusz Szpila; 27 October 1920 – 6 March 2002) was a Polish American geodesist who worked with the U.S. Air Force and later the National Geodetic Survey to adapt three-dimensional adjustment techniques to NAD 83. [1]
The solutions to both problems in plane geometry reduce to simple trigonometry and are valid for small areas on Earth's surface; on a sphere, solutions become significantly more complex as, for example, in the inverse problem, the azimuths differ going between the two end points along the arc of the connecting great circle.
"Nearly antipodal points" which describes the problems of failure to converge or slow convergence for the inverse method. This includes pointers to Vincenty's efforts to correct these problems. I also include a plug for my method of solving the inverse problem via Newton's method because
An inverse problem in science is the process of calculating from a set of observations the causal factors that produced them: for example, calculating an image in X-ray computed tomography, source reconstruction in acoustics, or calculating the density of the Earth from measurements of its gravity field. It is called an inverse problem because ...
Hansen's problem; Snellius–Pothenot problem; Great-circle distance – how to find that distance if one knows the latitude and longitude. Resection (orientation) Vincenty's formulae; Geographic distance; Triangulation in three dimensions