Geodesic arc length on the reference ellipsoid to second-order terms in the flattening

Abstract

Some known expansions of geodesic arc length on the reference ellipsoid to first order terms in the flattening are embodied in the Andoyer-Lambert formulas which are used extensively in Loran computations. The only known published derivation is by Andoyer, and the extension through his method to higher-order terms in the flattening is not apparent. Independent proofs of these approximations were devised from the great elliptic arc approximation to the geodesic, from modified differential equations of the geodesic length on the reference ellipsoid, and from identification in a paper published by A. R. Forsyth in 1895. In addition, after two errors in Forsyth's derivation of the second order term in the flattening had been detected and corrected, formulas were developed in terms of new parameters which give azimuths within a second of arc and distances within a meter for geodetic lines up to at least 10 million meters. A summary is presented of the results of computations for 17 known geodetic lines in several latitudes and azimuths.