His unpublished report (1975b) mentions the use of a Wang 720 desk calculator, which had only a few kilobytes of memory.
Legendre showed that an ellipsoidal geodesic can be exactly mapped to a great circle on the auxiliary sphere by mapping the geographic latitude to reduced latitude and setting the azimuth of the great circle equal to that of the geodesic.
Bessel and Helmert gave rapidly converging series for these integrals, which allow the geodesic to be computed with arbitrary accuracy.
The expressions were put in Horner (or nested) form, since this allows polynomials to be evaluated using only a single temporary register.
Define the following notation: Given the coordinates of the two points (Φ1, L1) and (Φ2, L2), the inverse problem finds the azimuths α1, α2 and the ellipsoidal distance s. Calculate U1, U2 and L, and set initial value of λ = L. Then iteratively evaluate the following equations until λ converges: When λ has converged to the desired degree of accuracy (10−12 corresponds to approximately 0.06 mm), evaluate the following: Between two nearly antipodal points, the iterative formula may fail to converge; this will occur when the first guess at λ as computed by the equation above is greater than π in absolute value.
[clarification needed] In his letter to Survey Review in 1976, Vincenty suggested replacing his series expressions for A and B with simpler formulas using Helmert's expansion parameter k1: where As noted above, the iterative solution to the inverse problem fails to converge or converges slowly for nearly antipodal points.
An example of an incorrect result is provided by the NGS online utility, which returns a distance that is about 5 km too long.
In an unpublished report, Vincenty (1975b) gave an alternative iterative scheme to handle such cases.