Solving the Direct and Inverse Geodetic Problems on the Ellipsoid by Numerical Integration
Publication: Journal of Surveying Engineering
Volume 138, Issue 1
Abstract
Taking advantage of numerical integration, we solve the direct and inverse geodetic problems on the ellipsoid. In general, the solutions are composed of a strict solution for the sphere plus a correction to the ellipsoid determined by numerical integration. Primarily the solutions are integrals along the geodesic with respect to the reduced latitude or azimuth, but these techniques either have problems when the integral passes a vertex (i.e., point with maximum/minimum latitude of the arc) or a singularity at the equator. These problems are eliminated when using Bessel’s idea of integration along the geocentric angle of the great circle of an auxiliary sphere. Hence, this is the preferred method. The solutions are validated by some numerical comparisons to Vincenty’s iterative formulas, showing agreements to within of geodesic length (or 3.1 mm) and as seconds of azimuth and position for baselines in the range of 19,000 km.
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Acknowledgments
We acknowledge the detailed comments by two unknown reviewers on a preliminary version of the manuscript.
References
Bessel, F. W. (1826). “Über die berechnung der geographischen längen und breiten aus geodätischen Vermessungen.” Astron. Nachr., 4(86), 241–254.
Heck, B. (1987). Rechenverfahren und Auswertemodelle der Landesvermessung, Herbert Wichmann Verlag, Karlsruhe, Germany.
Kivioja, L. A. (1971). “Computation of geodetic direct and indirect problems by computers accumulating increments from geodetic line elements.” Bull. Geod., 99(1), 55–63.
Klotz, J. (1991). “Eine analytische Lösung kanonischer Gleichungen der geodätischen Linien zur Transformation ellipsoidischer Flächenkoordinaten.” Deutsche Geod. Kommission Ser. C, No. 385, Bavarian Academy of Sciences, Munich.
Rainsford, H. F. (1955). “Long geodesics on the ellipsoid.” Bull. Geod., 37(1), 12–22.
Schmidt, H. (1999). “Lösung der geodätischen Hauptaufgaben auf dem Rotationsellipsoid mittels numerischer Integration.” Z. Vermessungsw. 124, 121–128.
Schmidt, H. (2006). “Note on Lars E. Sjöberg: New solutions to the direct and indirect geodetic problems on the ellipsoid.” Z. Vermessungsw., 131, 153–154.
Sjöberg, L. E. (2006a). “New solutions to the direct and indirect geodetic problems on the ellipsoid.” Zeitschrift fuer Vermessungswesen, 131, 35–39.
Sjöberg, L. E. (2006b). “Comment to H. Schmidt’s remarks on Sjöberg, zfv 1/2006, 35-39.” Z. Vermessungsw., 131, 155.
Sjöberg, L. E. (2006c). “Determination of areas on the plane, sphere and ellipsoid.” Surv. Rev., 38, 583–593.
Sjöberg, L. E. (2007). “Precise determination of the Clairaut constant in ellipsoidal geodesy.” Surv. Rev., 39(303), 81–86.
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 formula.” J. Surv. Eng., 131(1), 20–26.
Vincenty, T. (1975). “Direct and inverse solutions of geodesics on the ellipsoid with application of nested equations.” Surv.Rev., 23(176), 88–93.
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© 2012 American Society of Civil Engineers.
History
Received: Nov 30, 2010
Accepted: Jun 8, 2011
Published online: Jan 17, 2012
Published in print: Feb 1, 2012
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