Hostname: page-component-8448b6f56d-sxzjt Total loading time: 0 Render date: 2024-04-23T20:37:40.962Z Has data issue: false hasContentIssue false

An Algorithm for the Inverse Solution of Geodesic Sailing without Auxiliary Sphere

Published online by Cambridge University Press:  11 April 2014

Wei-Kuo Tseng*
(Department of Merchant Marine, National Taiwan Ocean University)


An innovative algorithm to determine the inverse solution of a geodesic with the vertex or Clairaut constant located between two points on a spheroid is presented. This solution to the inverse problem will be useful for solving problems in navigation as well as geodesy. The algorithm to be described derives from a series expansion that replaces integrals for distance and longitude, while avoiding reliance on trigonometric functions. In addition, these series expansions are economical in terms of computational cost. For end points located at each side of a vertex, certain numerical difficulties arise. A finite difference method together with an innovative method of iteration that approximates Newton's method is presented which overcomes these shortcomings encountered for nearly antipodal regions. The method provided here, which does not involve an auxiliary sphere, was aided by the Computer Algebra System (CAS) that can yield arbitrarily truncated series suitable to the users accuracy objectives and which are limited only by machine precisions.

Research Article
Copyright © The Royal Institute of Navigation 2014 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)



ArcGIS Resources. (2014). Calculating Geodesic Distance Between Points. Accessed 13 January 2014.Google Scholar
Bessel, F.W. (1826). Uber die Berechnung der Geographischen Langen und Breiten ausgeodatischen Vermessungen. (On the computation of geographical longitude and latitude grom geodetic measurements), Astronomische Nachrichten (Astronomical Notes), 4 (86), 241254.Google Scholar
Bowring, B.R. (1983). The geodesic inverse problem. Bulletin Geodesique, 57(2), 109120.Google Scholar
Bowring, B.R. (1984). Note on the geodesic inverse problem, Bulletin Geodesique, 58, 543.Google Scholar
Clairaut, A.C. (1735). Détermination géometrique de la perpendiculaireà la méridienne tracée par M Cassini. Mém de l'Acad Roy des Sciences de Paris 1733:406–416.Google Scholar
Jank, W. and Kivioja, L.A. (1980). Solution of the direct and inverse problems on reference ellipsoids by point-by-point integration using programmable pocket calculators. Surveying and Mapping, 15(3), 325337.Google Scholar
Helmert, F.R. (1880). Die matematischen und physicalischen Theorien der höheren Geodäsie, Part 1. B G Teubner, Leipzig.Google Scholar
Hipparchus® Tutorial and Programmer's Guide. (2004). Chapter 9: Geographics. Accessed 13 January 2014.Google Scholar
IBM DB2 Universal Database 9.1. (2012). Spatial functions supported by DB2 Geodetic Data Management Feature. Accessed 13 January 2014.Google Scholar
Kivioja, L.A. (1971). Computation of geodetic direct and indirect problems by computers accumulating increments from geodetic line elements. Bulletin Geodesique, 99, 5563.CrossRefGoogle Scholar
Karney, C.F.F. (2013). Algorithms for geodesics. Journal of Geodesy 87 (1), 43–42.Google Scholar
Microsoft SQL server. (2014). Spatial Data Types Overview Accessed 13 January 2014.Google Scholar
Oracle® Spatial Developer's Guide. (2013). Coordinate Systems (Spatial Reference Systems). Accessed 7 January 2014.Google Scholar
Rainsford, H.F. (1955) Long geodesics on the ellipsoid. Bulletin Geodesique, 37, 1222.Google Scholar
Rapp, R.H. (1993). Geometric Geodesy Part II. The Ohio State University.Google Scholar
Saito, T. (1970). The computation of long geodesics on the ellipsoid by non-series expanding procedure. Bulletin Geodesique, 98, 341374.Google Scholar
Saito, T. (1979). The computation of long geodesics on the ellipsoid through Gaussian quadrature. Bulletin Geodesique, 53(2), 165177.Google Scholar
Schmidt, H. (2006). Note on Lars E. Sjöberg: New solutions to the direct and indirect geodetic problems on the ellipsoid, ZfV, 1/2006, 3539.Google Scholar
Sjöberg, L.E. (2007). Precise determination of the Clairaut constant in ellipsoidal geodesy. Survey Review, 39, 8186.Google Scholar
Sjöberg, L.E. (2012). Solutions to the ellipsoidal Clairaut constant and the inverse geodetic problem by numerical integration. Journal of Geodetic Science, 2(3), 162171.Google Scholar
Thomas, C.M. and Featherstone, W.E. (2005). Validation of Vincenty's formulas for the geodesic using a new fourth-order extension of Kivioja's formual. Journal of Surveying Engineering, 131(1), 2026.Google Scholar
Tseng, W.K., Guo, J.L., Liu, C.P. and Wu, C.T. (2012a). The vector solutions for the great ellipse on the spheroid. Journal of Applied Geodesy, 6(2), 103109.Google Scholar
Tseng, W.K., Earle, M.A., and Guo, J.L. (2012b). Direct and Inverse Solutions with Geodetic Latitude in Terms of Longitude for Rhumb Line Sailing. Journal of Navigation, 65, 549559.Google Scholar
Tseng, W.K., Guo, J.L., Liu, C.P. (2013). A Comparison of Great Circle, Great Ellipse, and Geodesic Sailing. Journal of Marine Science and Technology, 21(3), 287299.Google Scholar
Vincenty, T. (1975a). Direct and inverse solutions of geodesics on the ellipsoid with application of nested equations. Survey Review, 23(176), 8893 [addendum: Surv Rev 23(180):294 (1976)].Google Scholar
Vincenty, T. (1975b). Geodetic inverse solution between antipodal points. Geographic library. Accessed 7 January 2014.Google Scholar