Background
Vincenty's goal was to express existing algorithms for the direct and inverse geodesic problem in a form that minimized the program length (see the first sentence of his paper). His unpublished report (1975b) mentions the use of a Wang 720 desk calculator which had only a few kilobytes of memory. To obtain good accuracy for long lines, the solution uses the classical solution of Legendre (1806), Bessel (1825), and Helmert (1880) based on the auxiliary sphere. (Vincenty relied on formulation of this method given by Rainsford, 1955.) 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. The longitude on the ellipsoid and the distance along the geodesic are then given in terms of the longitude on the sphere and the arc length along the great circle by simple integrals. Bessel and Helmert gave rapidly converging series for these integrals which allow the geodesic to be computed with arbitrary accuracy.
In order to minimize the program size, Vincenty took these series, re-expanded them using the first term of each series as the small parameter, and truncated them to order ƒ3. This resulted in compact expressions for the longitude and distance integrals. The expressions were put in Horner (or nested) form, since this allows polynomials to be evaluated using only a single temporary register. Finally, simple iterative techniques were used to solve the implicit equations in the direct and inverse methods; even though these are slow (and in the case of the inverse method it sometimes does not converge), they result in the least increase in code size.
Read more about this topic: Vincenty's Formulae
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