The geodesic inverse problem

Abstract

The first sentence of section (2) should read;For equatorial lines a 1 = 90° exactly, 'Y = y = 1 , and r = 1-'Y 2 = 0.However, I feel that the mathematical development given by the section would be well applicable to all cases, including the case a 1 = 0 exactly, but clearly the real need was for the special case of the line on the equator.

So, What is Actually the Distance from the Equator to the Pole? – Overview of the Meridian Distance Approximations

“…Despite the simplicity and fast convergence of Helmertʹs expansion the U.S. DMA adopted the fully expanded form of the Bessel series reported by Hinks in 1927. This expansion is important, despite the poorer convergence of series in n, because it is used in the definition of UTM [Bowring, 1983]…”

“…Equation 46can be written in the form of equation 47provided by Veis [Veis, 1992]   Another formula for the meridian arc length is equation 50, which is used by Bowring [Bowring, 1983] as the reference for the derivation of other formulas, employing polar coordinates and complex numbers. The basic difference of formula (50) from (47), (48) and (49) is that (50) uses the ellipsoid parameters (a, b), instead of the parameters (a, e) which are used in formulas (47), (48) and (49).…”

“…The different formulas and methods for the calculation of meridian arc distances, which have been initially evaluated and compared, are:  the proposed new formulas by Pallikaris,Tsoulos and Paradissis (53) and (54), with author's modification (55) ;  "Veis -Torge" formulas (formulas (47) and (48)) in various versions, according to the number of retained terms (1st version with up to M8 terms, 2nd version up to M6 terms, 3rd version up to M4 terms, 4th version up to M2 terms);  The Bowring [Bowring, 1983] formula (50);  The Bowring [Bowring, 1985] formula (51);…”

“…It is not surprise that they correspond to the results presented in Table 1. Deakin, 2010 (44) 10,001,965.72931270 10,001,965.72922300 Bomford, 1985 (49) 10,001,965.72931360 AMN, 1987 (49) 10,001,965.72952860 Bowring, 1983 (50) 10,001,965.72931270 Pallikaris, et al, 2009 (53) 10,001,965.72590000 Weintrit, 2013 (55) 10,001,965.72931270…”