1 Penev P.D.
2 Peneva E.P.

University of Architecture, Civil Engineering and Geodesy

The authors propose to derive the formulas given in [1, 2] for determining the height and latitude based on the Cartesian rectangular coordinates X, Y, Z, giving an accuracy for the geodetic height H of 1 mm for heights up to 50 km and for geodetic latitude B of 0,0001 arc seconds for H < 10 km. The formulas proposed in [1, 2] apply to all values of latitude and longitude (B and L). In [3], we propose two new formulas for H and B. In this paper, it is shown that the formulas proposed in [3] apply to points of ellipsoid surface and points with geodetic latitude of 0° and 90°. For the same formulas proposed in [3], the corrections are derived to ensure an accuracy of H of 1 mm at H ≤ 10 km, which apply to all values of B and L. Basing on the presented geometric conclusions, calculations and analyzes, a new solution for H and B respectively is proposed for given X, Y, Z, which provides an accuracy for H less than 1 mm for H ≤ 100 km and for B of 0,0001 arc seconds for H ≤ 50 km.
The authors' special thanks to candidate of technical sciences V.V Popadyov for his attention to the work and valuable suggestions.
1.   Balandin V.N., Bryn M.Ya., Imshenetskiy S.P., Matveev A.Yu., Yuskevich A.V. (2006) Algorithm for calculating the geodetic height by spatial rectangular coordinates. Geodesy and Cartography, 67(6), pp. 15–16.
2.   Balandin V.N., Bryn M.Ya., Menchikov I.V., Firsov Yu.G. (2012) On calculation of geodetic height on spatial rectangular coordinates. Geodezia i Kartografia, 73(2), pp. 9-11.
3.   Kureniov Y.P., Malik T.N. (2017) Definition of geodetic height namely by measured geocentric coordinates. Geodezia i Kartografia, (3), pp. 20-23. (In Russian). DOI: 10.22389/0016-7126-2017-921-3-20-23.
4.   Mustafin M.G., Tran Thanh Son (2018) Method for determining the normal heights from satellite data, taking into account the deviations of the plumb lines. Geodezia i Kartografia, 79(7), pp. 2-10. (In Russian). DOI: 10.22389/0016-7126-2018-937-7-2-10.
5.   Ogorodova L.V. (2017) Reliable and easy method for converting geodetic rectangular coordinates into curvilinear ones. Geodezia i Kartografia, (1), pp. 2-6. (In Russian). DOI: 10.22389/0016-7126-2017-919-1-2-6.
6.   Penev P., Peneva E. Preobrazovanie pryamougol'nyh geocentricheskih koordinat v geodezicheskie bez primeneniya iteracij. Izv. vuzov. Geodeziya i aehrofotos"emka, 2012, no. 3, pp. 34–38.
7.   Younes J.A., Mustafin M.G. (2018) The characteristics of transforming coordinates from the geocentric system WGS-84 for the Mercator projection under low latitudes conditions. Geodezia i Kartografia, 79(10), pp. 2-6. (In Russian). DOI: 10.22389/0016-7126-2018-940-10-2-6.
8.   Bowring B. (1985) The accuracy of geodetic latitude and height equations. Survey Review, no. 28 (218), pp. 200-206.
9.   Paul M. K. (1973) A note on computation of geodetic coordinates from geocentric (Cartesian) coordinates. Bulletin Géodésique, no. 108, pp. 135–139.
10.   Vermeille H. (2002) Computing geodetic coordinates from geocentric coordinates. Journal of Geodesy, no. 78, pp. 94–95.
Penev P.D., 
Peneva E.P., 
(2021) Determination of the geodetic latitude and height by spatial geocentric coordinates. Geodesy and cartography = Geodezia i Kartografia, 82(11), pp. 2-7. (In Russian). DOI: 10.22389/0016-7126-2021-977-11-2-7
Publication History
Received: 11.08.2020
Accepted: 29.09.2021
Published: 20.12.2021


2021 November DOI: