Abstract
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.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsReferences
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)
Awange JL, Grafarend EW, Fukuda Y, Takemoto S (2005) The application of commutative algebra to geodesy: two examples. J Geod 79:93–102
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 82:436–439
Benning W (1974) Der kürzeste Abstand eines in rechtwinkligen Koordinaten gegebenen Außenpunktes vom Ellipsoid. Allgemeine Vermessungs-Nachrichten 81:429–433
Benning W (1987) Iterative ellipsoidische Lotfußpunktberechung. Allgemeine Vermessungs-Nachrichten 94:256–260
Borkowski KM (1987) Transformation of geocentric to geodetic coordinates without approximation. Astrophys Space Sci 139:1–4
Borkowski KM (1989) Accurate algorithm to transform geocentric to geodetic coordinates. Bull Geod 63:50–56
Bowring BR (1976) Transformation from spatial to geographical coordinates. Surv Rev 23:323–327
Bowring BR (1985) The accuracy of geodetic latitude and height equations. Surv Rev 28:202–206
Croceto N (1993) Point projection of topographic surface onto the reference ellipsoid of revolution in geocentric Cartesian coordinates. Surv Rev 32:233–238
Fitzgibbon A, Pilu M, Fisher RB (1999) Direct least squares fitting of ellipses. IEEE Trans Pattern Anal Mach Intell 21:476–480
Fotiou A (1998) A pair of closed expressions to transform geocentric to geodetic coordinates. Zeitschrift für Vermessungswesen 123:133–135
Fröhlich H, Hansen HH (1976) Zur Lotfußpunktrechnung bei rotationsellipsoidischer Bezugsfläche. Allgemeine Vermessungs-Nachrichten 83:175–179
Fukushima T (1999) Fast transform from geocentric to geodetic coordinates. J Geod 73:603–610
Gander W, Golub GH, Strebel R (1994) Least-squares fitting of circles and ellipses. BIT No 43:558–578
Grafarend EW (2000) Gaußsche flächennormale Koordinaten im Geometrie- und Schwereraum. Erste Teil: Flächennormale Ellipsoidkoordinaten. Zeitschrift für Vermessungswesen 125:136–139
Grafarend EW (2000) Gaußsche flächennormale Koordinaten im Geometrie- und Schwereraum. Erste Teil: Flächennormale Ellipsoidkoordinaten. Zeitschrift für Vermessungswesen 125:136–139
Grafarend EW, Ardalan A (1999) World geodetic datum 2000. J Geod 73:611–623
Grafarend EW, Lohse P (1991) The minimal distance mapping of the topographic surface onto the (reference) ellipsoid of revolution. Manuscripta Geodaetica 16:92–110
Grafarend EW, Syffus R, You RJ (1995) Projective heights in geometry and gravity space. Allgemeine Vermessungs-Nachrichten 102:382–402
Heck B (1987) Rechenverfahren und Auswertemodelle der Landesvermessung. Wichmann Verlag, Karlsruhe
Heikkinen M (1982) Geschlossene Formeln zur Berechnung räumlicher geodätischer Koordinaten aus rechtwinkligen Koordinaten. Zeitschrift für Vermessungswesen 107:207–211
Heiskanen WA, Moritz H (1967) Physical geodesy. Freeman and Company, London
Hirvonen R, Moritz H (1963) Practical computation of gravity at high altitudes. Report No. 27, Institute of Geodesy, Photogrammetry and Cartography, Ohio State University, Ohio
Hofman-Wellenhof B, Lichtenegger H, Collins J (2001) Global positioning system: theory and practice, 5th edn. Springer, Wien
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 395–404
Lin KC, Wang J (1995) Transformation from geocentric to geodetic coordinates using Newton’s iteration. Bull Geod 69:300–303
Loskowski P (1991) Is Newton’s iteration faster than simple iteration for transformation between geocentric and geodetic coordinates? Bull Geod 65:14–17
Ozone MI (1985) Non-iterative solution of the φ equations. Surv Mapp 45:169–171
Paul MK (1973) A note on computation of geodetic coordinates from geocentric (Cartesian) coordinates. Bull Geod 108:135–139
Penev P (1978) The transformation of rectangular coordinates into geographical by closed formulas. Geo Map Photo 20:175–177
Pick M (1985) Closed formulae for transformation of Cartesian coordinates into a system of geodetic coordinates. Studia geoph et geod 29:653–666
Sjöberg LE (1999) An efficient iterative solution to transform rectangular geocentric coordinates to geodetic coordinates. Zeitschrift für Vermessungswesen 124:295–297
Soler T, Hothem LD (1989) Important parameters used in geodetic transformations. J Surv Eng 115:414–417
Sünkel H (1999) Ein nicht-iteratives Verfahren zur Transformation geodätischer Koordinaten. Öster. Zeitschrift für Vermessungswesen 64:29–33
Torge W (1991) Geodesy, 2nd edn. Walter de Gruyter, Berlin
Vanicek P, Krakiwski EJ (1982) Geodesy: the concepts. North-Holland Publishing Company, Amsterdam/New York/Oxford
Vincenty T (1978) Vergleich zweier Verfahren zur Berechnung der geodätischen Breite und Höhe aus rechtwinkligen koorninaten. Allgemeine Vermessungs-Nachrichten 85:269–270
Vincenty T (1980) Zur räumlich-ellipsoidischen Koordinaten-Transformation. Zeitschrift für Vermessungswesen 105:519–521
You RJ (2000) Transformation of Cartesian to geodetic coordinates without iterations. J Surv Eng 126:1–7
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2016 Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
Awange, J.L., Paláncz, B. (2016). Cartesian to Ellipsoidal Mapping. In: Geospatial Algebraic Computations. Springer, Cham. https://doi.org/10.1007/978-3-319-25465-4_14
Download citation
DOI: https://doi.org/10.1007/978-3-319-25465-4_14
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-25463-0
Online ISBN: 978-3-319-25465-4
eBook Packages: Earth and Environmental ScienceEarth and Environmental Science (R0)