Cartesian to Ellipsoidal Mapping

  • Joseph L. Awange
  • Béla Paláncz


In establishing a proper reference frame of geodetic point positioning, namely by the Global Positioning System (GPS) – the Global Problem Solver – we are in need to establish a proper model for the Topography of the Earth, the Moon, the Sun or planets. By the theory of equilibrium figures, we are informed that an ellipsoid, two-axes or three-axes is an excellent approximation of the Topography. For planets similar to the Earth the biaxial ellipsoid, also called “ellipsoid-of-revolution” is the best approximation.


Global Position System Reference Ellipsoid Ellipsoidal Height Quartic Polynomial Gauss Surface 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


  1. 17.
    Awange JL (2002) Groebner bases, multipolynomial resultants and the Gauss-Jacobi combinatorial algorithms-adjustment of nonlinear GPS/LPS observations. Ph.D. thesis, Department of Geodesy and GeoInformatics, Stuttgart University, Germany. Technical reports, Report Nr. 2002 (1)Google Scholar
  2. 42.
    Awange JL, Grafarend EW, Fukuda Y, Takemoto S (2005) The application of commutative algebra to geodesy: two examples. J Geod 79:93–102CrossRefGoogle Scholar
  3. 65.
    Bartelme N, Meissl P (1975) Ein einfaches, rasches und numerisch stabiles Verfahren zur Bestimmung des kürzesten Abstandes eines Punktes von einem sphäroidischen Rotationsellipsoid. Allgemeine Vermessungs-Nachrichten 82:436–439Google Scholar
  4. 77.
    Benning W (1974) Der kürzeste Abstand eines in rechtwinkligen Koordinaten gegebenen Außenpunktes vom Ellipsoid. Allgemeine Vermessungs-Nachrichten 81:429–433Google Scholar
  5. 78.
    Benning W (1987) Iterative ellipsoidische Lotfußpunktberechung. Allgemeine Vermessungs-Nachrichten 94:256–260Google Scholar
  6. 95.
    Borkowski KM (1987) Transformation of geocentric to geodetic coordinates without approximation. Astrophys Space Sci 139:1–4CrossRefGoogle Scholar
  7. 96.
    Borkowski KM (1989) Accurate algorithm to transform geocentric to geodetic coordinates. Bull Geod 63:50–56CrossRefGoogle Scholar
  8. 98.
    Bowring BR (1976) Transformation from spatial to geographical coordinates. Surv Rev 23:323–327CrossRefGoogle Scholar
  9. 99.
    Bowring BR (1985) The accuracy of geodetic latitude and height equations. Surv Rev 28:202–206CrossRefGoogle Scholar
  10. 138.
    Croceto N (1993) Point projection of topographic surface onto the reference ellipsoid of revolution in geocentric Cartesian coordinates. Surv Rev 32:233–238CrossRefGoogle Scholar
  11. 172.
    Fitzgibbon A, Pilu M, Fisher RB (1999) Direct least squares fitting of ellipses. IEEE Trans Pattern Anal Mach Intell 21:476–480CrossRefGoogle Scholar
  12. 175.
    Fotiou A (1998) A pair of closed expressions to transform geocentric to geodetic coordinates. Zeitschrift für Vermessungswesen 123:133–135Google Scholar
  13. 179.
    Fröhlich H, Hansen HH (1976) Zur Lotfußpunktrechnung bei rotationsellipsoidischer Bezugsfläche. Allgemeine Vermessungs-Nachrichten 83:175–179Google Scholar
  14. 181.
    Fukushima T (1999) Fast transform from geocentric to geodetic coordinates. J Geod 73:603–610CrossRefGoogle Scholar
  15. 185.
    Gander W, Golub GH, Strebel R (1994) Least-squares fitting of circles and ellipses. BIT No 43:558–578CrossRefGoogle Scholar
  16. 207.
    Grafarend EW (2000) Gaußsche flächennormale Koordinaten im Geometrie- und Schwereraum. Erste Teil: Flächennormale Ellipsoidkoordinaten. Zeitschrift für Vermessungswesen 125:136–139Google Scholar
  17. 208.
    Grafarend EW (2000) Gaußsche flächennormale Koordinaten im Geometrie- und Schwereraum. Erste Teil: Flächennormale Ellipsoidkoordinaten. Zeitschrift für Vermessungswesen 125:136–139Google Scholar
  18. 209.
    Grafarend EW, Ardalan A (1999) World geodetic datum 2000. J Geod 73:611–623CrossRefGoogle Scholar
  19. 214.
    Grafarend EW, Lohse P (1991) The minimal distance mapping of the topographic surface onto the (reference) ellipsoid of revolution. Manuscripta Geodaetica 16:92–110Google Scholar
  20. 232.
    Grafarend EW, Syffus R, You RJ (1995) Projective heights in geometry and gravity space. Allgemeine Vermessungs-Nachrichten 102:382–402Google Scholar
  21. 265.
    Heck B (1987) Rechenverfahren und Auswertemodelle der Landesvermessung. Wichmann Verlag, KarlsruheGoogle Scholar
  22. 266.
    Heikkinen M (1982) Geschlossene Formeln zur Berechnung räumlicher geodätischer Koordinaten aus rechtwinkligen Koordinaten. Zeitschrift für Vermessungswesen 107:207–211Google Scholar
  23. 268.
    Heiskanen WA, Moritz H (1967) Physical geodesy. Freeman and Company, LondonGoogle Scholar
  24. 269.
    Hirvonen R, Moritz H (1963) Practical computation of gravity at high altitudes. Report No. 27, Institute of Geodesy, Photogrammetry and Cartography, Ohio State University, OhioGoogle Scholar
  25. 275.
    Hofman-Wellenhof B, Lichtenegger H, Collins J (2001) Global positioning system: theory and practice, 5th edn. Springer, WienCrossRefGoogle Scholar
  26. 321.
    Lapaine M (1990) A new direct solution of the transformation problem of Cartesian into ellipsoidal coordinates. In: Rapp RH, Sanso F (eds) Determination of the geoid: present and future. Springer, New York, pp 395–404Google Scholar
  27. 340.
    Lin KC, Wang J (1995) Transformation from geocentric to geodetic coordinates using Newton’s iteration. Bull Geod 69:300–303CrossRefGoogle Scholar
  28. 344.
    Loskowski P (1991) Is Newton’s iteration faster than simple iteration for transformation between geocentric and geodetic coordinates? Bull Geod 65:14–17Google Scholar
  29. 393.
    Ozone MI (1985) Non-iterative solution of the φ equations. Surv Mapp 45:169–171Google Scholar
  30. 405.
    Paul MK (1973) A note on computation of geodetic coordinates from geocentric (Cartesian) coordinates. Bull Geod 108:135–139CrossRefGoogle Scholar
  31. 407.
    Penev P (1978) The transformation of rectangular coordinates into geographical by closed formulas. Geo Map Photo 20:175–177Google Scholar
  32. 410.
    Pick M (1985) Closed formulae for transformation of Cartesian coordinates into a system of geodetic coordinates. Studia geoph et geod 29:653–666CrossRefGoogle Scholar
  33. 467.
    Sjöberg LE (1999) An efficient iterative solution to transform rectangular geocentric coordinates to geodetic coordinates. Zeitschrift für Vermessungswesen 124:295–297Google Scholar
  34. 468.
    Soler T, Hothem LD (1989) Important parameters used in geodetic transformations. J Surv Eng 115:414–417CrossRefGoogle Scholar
  35. 479.
    Sünkel H (1999) Ein nicht-iteratives Verfahren zur Transformation geodätischer Koordinaten. Öster. Zeitschrift für Vermessungswesen 64:29–33Google Scholar
  36. 489.
    Torge W (1991) Geodesy, 2nd edn. Walter de Gruyter, BerlinCrossRefGoogle Scholar
  37. 496.
    Vanicek P, Krakiwski EJ (1982) Geodesy: the concepts. North-Holland Publishing Company, Amsterdam/New York/OxfordGoogle Scholar
  38. 501.
    Vincenty T (1978) Vergleich zweier Verfahren zur Berechnung der geodätischen Breite und Höhe aus rechtwinkligen koorninaten. Allgemeine Vermessungs-Nachrichten 85:269–270Google Scholar
  39. 502.
    Vincenty T (1980) Zur räumlich-ellipsoidischen Koordinaten-Transformation. Zeitschrift für Vermessungswesen 105:519–521Google Scholar
  40. 540.
    You RJ (2000) Transformation of Cartesian to geodetic coordinates without iterations. J Surv Eng 126:1–7CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  • Joseph L. Awange
    • 1
  • Béla Paláncz
    • 2
  1. 1.Curtin UniversityPerthAustralia
  2. 2.Budapest University of Technology and EconomicsBudapestHungary

Personalised recommendations