Vincenty's Formulae - Nearly Antipodal Points

Nearly Antipodal Points

As noted above, the iterative solution to the inverse problem fails to converge or converges slowly for nearly antipodal points. An example of slow convergence is (φ₁, L₁) = (0°, 0°) and (φ₂, L₂) = (0.5°, 179.5°) for the WGS84 ellipsoid. This requires about 130 iterations to give a result accurate to 1 mm. Depending on how the inverse method is implemented, the algorithm might return the correct result (19936288.579 m), an incorrect result, or an error indicator. An example of an incorrect result is provided by the NGS online utility which returns a distance which is about 5 km too long. Vincenty suggested a method of accelerating the convergence in such cases (Rapp, 1973).

An example of a failure of the inverse method to converge is (φ₁, L₁) = (0°, 0°) and (φ₂, L₂) = (0.5°, 179.7°) for the WGS84 ellipsoid. In an unpublished report, Vincenty (1975b) gave an alternative iterative scheme to handle such cases. This converges to the correct result 19944127.421 m after about 60 iterations; however, in other cases many thousands of iterations are required.

Newton's method has been successfully used to give rapid convergence for all pairs of input points (Karney, 2013).