Jump to ContentJump to Main Navigation
Show Summary Details
More options …

Journal of Geodetic Science

Editor-in-Chief: Eshagh, Mehdi

1 Issue per year

Open Access
Online
ISSN
2081-9943
See all formats and pricing
More options …

Various parameterizations of “latitude” equation – Cartesian to geodetic coordinates transformation

M. Ligas
  • Corresponding author
  • AGH University of Science and Technology, Faculty of Mining Surveying and Environmental Engineering, Department of Geomatics, al. A. Mickiewicza 30, 30-059 Krakow, Poland
  • Email
  • Other articles by this author:
  • De Gruyter OnlineGoogle Scholar
Published Online: 2013-09-07 | DOI: https://doi.org/10.2478/jogs-2013-0012

Abstract

The paper presents a solution to one of the basic problems of computational geodesy - conversion between Cartesian and geodetic coordinates on a biaxial ellipsoid. The solution is based on what is known in the literature as “latitude equation”. The equation is presented in three different parameterizations commonly used in geodesy - geodetic, parametric (reduced) and geocentric latitudes. Although the resulting equations may be derived in many ways, here, we present a very elegant one based on vectors orthogonality. As the “original latitude equations” are trigonometric ones, their representation has been changed into an irrational form after Fukushima (1999, 2006). Furthermore, in order to avoid division operations we have followed Fukushima’s strategy again and rewritten the equations in a fractional form (a pair of iterative formulas). The resulting formulas involving parametric latitude are essentially the same as those introduced by Fukushima (2006) (considered the most efficient today). All the resulting variants are solved with Newton’s second-order and Halley’s third-order formulas. It turns out that all parameterizations of the “latitude equation” show a comparable level of performance.

Keywords : biaxial ellipsoid; Cartesian and geodetic coordinates; latitude equation

  • Borkowski K.M., 1987, Transformation of geocentric to geodetic coordinates without approximations, Astrophy. Space Sci., 139, 1-4Google Scholar

  • Borkowski K.M., 1989, Accurate algorithm to transform geocentric to geodetic coordinates, Bull. Geod., 63, 50-56Google Scholar

  • Bowring B.R., 1976, Transformation from spatial to geographical coordinates, Surv. Rev., 23, 323-327Google Scholar

  • Feltens J., 2008, Vector methods to compute azimuth, elevation, ellipsoidal normal, and Cartesian (X,Y,Z) to geodetic (ϕ,λ,h) transformation, J. Geod., 82, 493-504Web of ScienceGoogle Scholar

  • Fukushima T., 1999, Fast transform from geocentric to geodetic coordinates, J. Geod., 73, 603-610.Google Scholar

  • Fukushima T., 2006, Transformation from Cartesian to geodetic coordinates accelerated by Halley’s method, J. Geod., 79, 689-693Google Scholar

  • Hedgley D.R., 1976, An exact transformation from geocentric to geodetic coordinates for nonzero altitudes, NASA TR R - 458, WashingtonGoogle Scholar

  • Heiskanen W. A. and Moritz H., 1967, Physical Geodesy, W. H. Freeman and Company, San Francisco Householder A. S., 1970, The numerical treatment of a single nonlinear equation, McGraw - Hill, New YorkGoogle Scholar

  • Kincaid D. and Cheney W., 1991, Numerical Analysis, Brooks/Cole Publishing Company, Pacific Grove, California Lin K.C. and Wang J., 1995, Transformation from geocentric to geodetic coordinates using Newton’s iteration, Bull. Geod., 69, 300-303.Google Scholar

  • Nurnberg R., 2006, Distance from a point to an ellipse/ellipsoid, http://www2.imperial.ac.uk/~rn/distance2ellipse.pdfGoogle Scholar

  • Vermeille H., 2002, Direct transformation from geocentric coordinates to geodetic coordinates. J. Geod., 76, 451-454Web of ScienceGoogle Scholar

  • Vermeille H., 2004,Computing geodetic coordinates from geocentric coordinates. J. Geod., 78, 94-95Web of ScienceGoogle Scholar

  • Zhang C. D., Hsu H. T., Wu X. P., Li S. S., Wang Q. B., Chai A. Z. and Du L., 2005, An alternative algebraic algorithm to transform Cartesian to geodetic coordinates, J. Geod., 79, 413-420Google Scholar

  • Google Scholar

About the article

Published Online: 2013-09-07

Published in Print: 2013-09-01


Citation Information: Journal of Geodetic Science, Volume 3, Issue 2, Pages 87–110, ISSN (Print) 2081-9943, DOI: https://doi.org/10.2478/jogs-2013-0012.

Export Citation

This content is open access.

Comments (0)

Please log in or register to comment.
Log in