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Solution for azimuth of the geodesic in near antipodal situations with special reference to the behaviour of lines for which the azimuth is in the region of 90°

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Abstract

To deal specifically with the case of a nearly antipodal geodesic of given terminal positions, an equation in azimuth is produced for solution by the Newton-Raphson Method of successive approximation. When the azimuth is in the region of 90° it is found that an approach is best made by a development in series when the aim is precision calculation. Degeneracy is examined theoretically and graphically.

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Abbreviations

e:

eccentricity of the spheroid

f:

flattening of the spheroid

φ1, φ2 :

geographical latitudes of the ends of the lineP 1 ,P 2 respectively

U 1 ,U 2 :

parametric latitudes corresponding to φ1, φ2 respectively

L :

difference in geographical longitude between the ends of the lineP 1 P 2 . Also as longitude for a point at parametric latitude U on a geodesic for terms ofdU/dL, etc...

a 1,a 2 :

azimuths at the ends of the lineP 1 ,P 2 respectively

a :

azimuth of the line at the equator

λ:

spherical difference in longitude corresponding toL

σ:

spherical central angle

θ:

90°−a

s1 :

sinU1

c1 :

cosU1

s2 :

sinU2

c2 :

cosU2

Reference

  • H. F. RAINSFORD: Long geodesics on the ellipsoid., B. Géodésique, 37, 12–22. 1955.

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Authors and Affiliations

  1. Directorate of Overseas Surveys, Surrey, UK

    B. R. Bowring

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  1. B. R. Bowring
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Bowring, B.R. Solution for azimuth of the geodesic in near antipodal situations with special reference to the behaviour of lines for which the azimuth is in the region of 90°. Bull. Geodesique 51, 17–32 (1977). https://doi.org/10.1007/BF02521538

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  • DOI: https://doi.org/10.1007/BF02521538

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