Methods for Transformation of Rectangular Spatial Coordinates to Geodetic Coordinates

  • Authors

    • Pavel Aleksandrovich Medvedev
    • Leonid Vasilevich Bykov
    • Vasiliy Leonidovich Bykov
    • Marina Vladimirovna Novorodskaya
    • Svetlana Ivanovna Sherstneva
    2018-12-03
    https://doi.org/10.14419/ijet.v7i4.38.27758
  • transformation of coordinates, rectangular and geodetic coordinates, latitude, longitude, algorithms, Newton method.

  • Abstract

    The article gives a brief analysis of methods and algorithms for the transformation of spatial rectangular coordinates to curvilinear coordinates - geodetic latitude, geodetic longitude, geodetic height. Two algorithms for solving the equation for determining longitude are considered. Three formulas used to calculate the height are analyzed, with an estimate of their errors due to the approximate latitude. The shortcomings of mathematical solutions to these problems are revealed. A study of different approaches and methods for solving the transcendental equation for determining the latitude, based on the theory of separation of the root of the equation, is performed. Using this technique, iterative processes were performed to calculate the reduced latitude , using trigonometric identities, by introducing an auxiliary angle and transforming it to an algebraic quartic equation, which Borkowski solves by the Ferrari's method. The determination of the root isolation interval allowed using the chord method (proportional parts) to determine the latitude. In all cases, estimates of the convergence of the iterative processes that facilitate the comparative analysis of the proposed solutions are obtained. By further decreasing the separation interval of the root, the accuracy of the non-iterative determination of the latitude is improved by the Newton method.

     

     

  • References

    1. [1] Balandin VN (1988), Preobrazovanie prostranstvennykh pryamougolnykh koordinat v geodezicheskie [Transformation of spatial rectangular coordinates into geodesics]. Geodeziya i kartografiya, 1, 42-43.

      [2] Vermeille H (2002). Direct transformation from geocentric to geodetic coordinates. Journal of Geodesy, 76, 451-454.

      [3] Penev P, Peneva E. (2012), Preobrazovanie pryamougolnykh geotsentricheskikh koordinat v geodezicheskie bez primeneniya iteratsiy [Transformation of rectangular geocentric coordinates into geodesics without application of iterations]. Izvestiya vuzov. Geodeziya i aerofotosiemka, 3, 34-38.

      [4] Ogorodova LV (2014), Prostoy i nadezhnyy sposob vychisleniya geodezicheskoy shiroty i vysoty tochek poverkhnosti Zemli po pryamougol'nym koordinatam [A simple and reliable way of calculating the geodetic latitude and the height of the Earth's surface points along rectangular coordinates]. Izvestiya vuzov. Geodeziya i aerofotosiemka, 2, 63-66.

      [5] Medvedev PA (2016), Vysokotochnye algoritmy dlya pryamogo vychisleniya prostranstvennykh geodezicheskikh koordinat po pryamougol'nym koordinatam [High-precision algorithms for direct calculation of spatial geodetic coordinates by rectangular coordinates]. Geodeziya i kartografiya, 11, 2-6.

      [6] Ozone MI (1985), A non-iterative solution of the equation. Surv. and Mapping, 45(2), 169-171.

      [7] Varao Romao MS (1987), Тransformacao de Coordenadas Cartesianas Tridimensionais em Geograficas por um proc esso directo. Rev. Inst. Geogr. e Cadastral, 7, 87-94.

      [8] Medvedev PA (2000), Analiz preobrazovaniy prostranstvennykh pryamougol'nykh koordinat v geodezicheskie [Analysis of transformations of spatial rectangular coordinates into geodesics]. Omsk, Russia: Izd-vo OmGAU.

      [9] Medvedev PA (1994), Issledovanie rekurrentnykh formul dlya vychisleniya shiroty pri perekhode ot prostranstvennykh pryamougol'nykh koordinat k geodezicheskim [Investigation of recurrent formulas for calculating latitude in the transition from spatial rectangular coordinates to geodesic]. Geodeziya i kartografiya, 1(6), 8-14; 2(7), 8-12; 3(8), 4-9.

      [10] Laping KA (1964), O preobrazovanii pryamougolnykh prostranstvennykh koordinat v geodezicheskie koordinaty [On the transformation of rectangular spatial coordinates into geodetic coordinates]. Geodeziya i kartografiya, 6, 33-35.

      [11] Laping KA (1962), Vychislenie koordinat i vysot tochek po izmerennym azimutam normal'nykh secheniy i uglam naklona khord na dvukh iskhodnykh punktakh [Calculation of coordinates and heights of points from the measured azimuths of normal sections and angles of slope of chords at two initial points]. Izvestiya vuzov. Geodeziya i aerofotosiemka, 1, 1-8.

      [12] Butkevich AV (1967) O perekhode ot prostranstvennykh pryamougol'nykh koordinat k geodezicheskim [On the transition from spatial rectangular coordinates to geodesics]. Geodeziya i kartografiya, 6, 20-22.

      [13] Izotov AA (1969), Preobrazovanie prostranstvennykh pryamougol'nykh koordinat v geodezicheskie koordinaty [Transformation of spatial rectangular coordinates into geodetic coordinates]. Geodeziya i kartografiya, 5, 6-7.

      [14] Ganshin VN (1970), Perekhod ot prostranstvennykh pryamougol'nykh koordinat k geodezicheskim [Transition from spatial rectangular coordinates to geodesic]. Geodeziya i kartografiya, 1, 17-19.

      [15] Bowring BR (1976), Transformation from spatial to geographical coordinates. Survey Review, 23(181), 323-327.

      [16] Mashimov MM (1991), Geodeziya. Teoreticheskaya geodeziya: spravochnoe posobie [Geodesy. Theoretical Geodesy: A Reference Manual]. Moscow, Russia: Nedra.

      [17] Morozov VP (1969), Kurs sferoidicheskoy geodezii [Course of spheroidal geodesy]. Moscow, USSR: Nedra.

      [18] Andreev M (1966), Preobrazovanie pryamougolnykh prostranstvennykh koordinat v geodezicheskie [Transformation of rectangular spatial coordinates into geodesics]. Geodeziya i kartografiya, 9, 12-14.

      [19] Bencini P (1968), Sul calcolo di transformazione delle coordinate dal sistema geografico. Boll. geod. sci. affini, 27(3), 357-367.

      [20] Urmaev MS (1989), Kosmicheskaya fotogrammetriya: uchebnik dlya vuzov [Space photogrammetry: a textbook for universities]. Moscow, USSR: Nedra.

      [21] Balandin VN, Bryn MYa, Menshikov IV, Firsov YuG (2012), K voprosu vychisleniya geodezicheskoy shiroty po prostranstvennym pryamougol'nym koordinatam [To the question of calculating geodetic latitude with respect to spatial rectangular coordinates]. Geodeziya i kartografiya, 2, 9-11.

      [22] Rinner K (1958), Geometrie mit Raumstrecken. Zeitschrift für Vermessungswesen, 83(3), 17-21.

      [23] Pick M (1967), Transformation of the spatial rectangular coordinates into the geodetic coordinates. Bull. Geod., 83, 21-26.

      [24] Bopp H, Krauss H, (1976). Der kürzeste Abstand eines Punktes von einem Rotationsellopsoid. Allg. Vermess.: Nachr., 83(10), 348-351.

      [25] Vincenty T (1978), Vergleich zweier verfahren zur Berechnung der geoodatischen Breite und Hohe aus rechtwinkligen Koordinaten. Allg. Vermess.: Nachr., 85(7), 269-270.

      [26] Butkevich AV (1963), Perekhod ot prostranstvennykh pryamougol'nykh koordinat k geodezicheskim [Transition from spatial rectangular coordinates to geodesic]. Geodeziya i kartografiya, 10, 12-17.

      [27] Paul MK (1973), A note on computation of geodetic coordinates from geocentric (cartesian) coordinates. Bull. Geod., 108, 135-139.

      [28] Penev P (1973), Zatvoreni formuli za transformatsiya na pravoglni prostranstveni koordinati v geografski [Closed Formulas for the Transformation of Rectangular Space Coordinates in Geography]. Izvest. Glavn. Upravl. Geod. i Kartogr., 4, 29-36.

      [29] Benning W (1974), Der kürzeste Abstand eines in rechtwinklungen Koordinaten gegebenen Auβenpunktes vom Ellipsoid. Allg. Vermess.: Nachr., 81(11), 429-433.

      [30] Frohlich H (1976), Zur Lotfuβ punktberechnung bei rotationsellipsoidischer Bezugsflache. Allg. Vermess.: Nachr., 83(5), 175-179.

      [31] Heikkinen M (1982), Geschlossene Formeln zur Berechnung räumlicher geodätischer Koordinaten aus rechtwinkligen Koordinaten. Zeitschrift bur Vermessungswesen, 107(5), 207-211.

      [32] Borkowski KM (1989), Accurate algorithms to transform geocentric to geodetic coordinates. Bull. Geod., 63(1), 50-56.

      [33] Fukushima T (1999), Fast transform from geocentric to geodetic coordinates. Journal of Geodesy, 73, 603-610.

      [34] Bowring BR (1985), The accuracy of geodetic latitude and height equations. Survey Review, 38, 202-206.

      [35] Medvedev PA (1995), Vychislenie shiroty metodom kasatelnykh pri perekhode ot prostranstvennykh pryamougol'nykh koordinat k geodezicheskim [Calculation of latitude by the method of tangents in the transition from spatial rectangular coordinates to geodesics]. Geodeziya i kartografiya, 12, 4-6.

  • Downloads

  • How to Cite

    Aleksandrovich Medvedev, P., Vasilevich Bykov, L., Leonidovich Bykov, V., Vladimirovna Novorodskaya, M., & Ivanovna Sherstneva, S. (2018). Methods for Transformation of Rectangular Spatial Coordinates to Geodetic Coordinates. International Journal of Engineering & Technology, 7(4.38), 1179-1186. https://doi.org/10.14419/ijet.v7i4.38.27758

    Received date: 2019-02-21

    Accepted date: 2019-02-21

    Published date: 2018-12-03