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Getchell’s method for conversion of Cartesian-geocentric to geodetic coordinates – Its properties and Newtonian alternative

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Published/Copyright: November 4, 2020
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Journal of Applied Geodesy
From the journal Journal of Applied Geodesy Volume 15 Issue 1

Abstract

The paper concentrates on the iterative Getchell’s method (formulated in 1972) and its alternative Newtonian implementation for conversion of Cartesian geocentric coordinates into geodetic coordinates. The same basic equation formulated in the Getchell’s method is used in both cases. The equation has a stable form in the whole range of argument (latitude) variation ⟨−π/2,π/2⟩. The original Getchell’s method (somehow “forgotten”) has a simple geometric interpretation and its applications turn out to be particularly effective. Many studies on iterative algorithms usually omit theoretical proofs of convergence replacing them with conclusions based on numerical examples. This paper presents theoretical proofs of algorithms convergence both for the Getchell’s method and the Newton procedure. The convergence parameter and numerical error of results were estimated in each case. Numerical tests were carried out for a set of points distributed on the Earth’s space, also for extreme h values. For typical practical applications of the Getchell’s method, sufficiently accurate results are obtained after 1–3 iterations, while in the Newton procedure already after one iteration, assuming the same numerical error and initial conditions. The accuracy of the geodetic coordinates determinations meets all practical requirements with some margin. For example an absolute numerical error for latitude is approx. 0.4·10−13 [rad] i. e. about 0.00026 mm in the length of the meridian arc. The proposed methods were compared with other methods (algorithms), including in terms of stability and non-singularity in the entire usable space of the Earth, but excluding the near geocenter, which has no practical significance. Both the modification of the Getchell method and its Newtonian alternative are very good determined in this area (in the Earth’s poles, the final solution is directly the starting value of iterative algorithms). The discussed algorithms were implemented in the form of procedures in DELPHI language.

Keywords: geocentric coordinates; geodetic coordinates; conversion; iterative formula; convergence of algorithms

Acknowledgment

The work has been elaborated under the statutory research in Department of Geodesy and Geotechnics at Rzeszów University of Technology. I thank the reviewers whose comments have contributed to the improvement of this work.

Appendix A Algorithm implementations

The Getchell’s algorithm, with the modified starting point, is written below in the form of the procedure in DELPHI procedure XYZ_BLH(var X, Y, Z, WB, WL, h: double; var idefect: integer); //------------------- comments ------------------ // Cartesian geocentric to geodetic coordinates // // X,Y,Z : input - geocentric cartesian coordinates // WB,WL : output - geodetic coordinates in [sec] // h : output - ellipsoidal height // a, ff : a - semi-major axis and flattening of an ellipsoid // idefect: if idefect > 0 -> error information //----------------------------------------------- var iter: integer; a,ff,S0,S1,re,b,w,ss,RMmax, r,d,v,ee,ee1,ro,BB,BB1,s,c,RN,RM,th,dB: double; label e,err_end; begin //------ GRS-80 parameters ------------------ a:=6378137.0; ff:=0.003352810681182317; b:=a*(1-ff); // semi-minor axis RMmax:=a*a/b; ee:=1-sqr(1-ff); //the first eccentricity squared ee1:=ee/(1-ee); //the second eccentricity squared //------------------------------------------- WB:=0; WL:=0; h:=0; ro:=648000/pi; r:=sqrt(X*X+Y*Y); d:=sqrt(r*r+Z*Z); if(d<150000)thenbegin idefect:=1; goto err_end; end; v:=1/d; th:=arcsin(Z/d); re:=r/sqrt(r*r/(a*a)+z*z/(b*b)); s:= Z/sqrt(sqr(r-re*ee)+Z*Z); BB:=arcsin(s); iter:=0; e: if(iter>10) thenbegin idefect:=2; goto err_end; end; BB1:=BB; ss:=s*s; w:=1/sqrt(1-ee*ss); RN:=a*w; S0:=ee*RN*v*s*sqrt(1-ss); //--- Begin of code exchanged between procedures dB:=arcsin(S0); BB:=th+dB; //----- end of code exchanged between procedures if(abs(BB-BB1)*RMmax > 0.000001) thenbegin k:=k+1; s:=sin(BB); goto e; end; WB:=BB*ro; //------- longitude --------------------- if(r<0.0001) then WL:=0 else WL:= arcsin(abs(Y)/r); if(X<=0)and(Y>0) then WL:=pi-WL else if(X>=0)and(Y<0) then WL:=-WL else if(X<0)and(Y<=0) then WL:=-(pi-WL); WL:=WL*ro; //------- height ------------------------ s:=sin(BB); c:=cos(BB); RN:=a/sqrt(1-ee*s*s); if(abs(c)>abs(s)) then h:=r/c-RN else h:=Z/s-RN*(1-ee); err_end:; end; // end of procedure For the Newton’s procedure, it is enough to replace the indicated code fragment with the following instructions string: //---- begin of code exchanged between procedures RM:=RN*(1-ee)*w*w; S1:=ee1*RM*v*(1-2*ss+ee*ss*ss); dB:=(BB-th-arcsin(S0))/(1-(S1/sqrt(1-S0*S0))); BB:=BB1-dB; //---- end of code exchanged between procedures The implemented algorithms for φ:

  1. Data and constants: X,Y,Z (point coordinates), a (semi-major halbaxe), f=(a−b)/a (flattening)

    • b:=a·(1−f) (semi-minor halbaxe),

    • ee:=e2, ee1:=(e′)2: the first – and second eccentricity squared,

    • RMmax=a2/b: maximal radius of curvature in meridian,

    • eps:=0.000001: length of small arc (in [m]) als numeric error,

  2. Input data and the start point (iteration index k=0)

    • k:=0,

    • r:=(X2+Y2)1/2,

    • d=(r2+Z2)1/2,v=1/d,

    • ϑ=arcsin(Z·v):geocentric latitude

    • p0=r·{1−ee/[(r/a)2+(Z/b)2]1/2},

    • s=Z/(Z2+p02)1/2,

    • φ0=arcsin(s).

  3. Next (k+1) iteration (alternative instruction for Getchell method and Newtonian alternative):

    • ss:=s·s,

    • w:=1/(1−ee·ss)1/2,RN:=a·w,

    • S0:=RN·v·ee·s·(1−ss)1/2,

    • For Getchell’s method: φk+1:=ϑ+arcsin(S0),

    • For Newtonian alternative:

      • RM:=RN·(1−ee)·w2,

      • f0:=φk−ϑ−arcsin(S0),

      • S1:=ee1·RM·v·(1−2·ss+ee·ss·ss),

      • f1:=1−S1/[1−S0·S0]1/2,

      • dφ:=f0/f1,

      • φk+1:=φk−dφ,

  4. If | φk+1−φk|·RMmax>0.000001 m then

    • k:=k+1,

    • s:=sin(φk+1),

    • goto step 2.

  5. φ∗:=φk+1 (result)

The procedures include the complete task of converting Cartesian coordinates to geodetic coordinates, thus also calculating the longitude (λ) and ellipsoidal height (h). This is done according to generally known formulas (data: φ,X,Y,Z):

(A1)λo=arcsin(|Y|/r)forr>0;r=(X2+Y2)1/2

(defect protection condition was adopted: if r<0.0001 then will assumed λo=0). The final formula for longitude would be expressed elementarily as:

(A2)λ=λoifX>0andY≥0π−λoelse ifX≤0andY>0−(π−λo)else ifX<0andY≤0−λoelse(i.e. ifX≥0andY<0)
(A3)h=r/cos(φ)−RNif|φ|<π/4Z/sin(φ)−RN·(1−e2)else(i.e.|φ|≥π/4)
Alternative formulas for h protect against numerical instability, see e. g. Fukushima [13].

References

[1] Bartelme, N. and 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.Search in Google Scholar

[2] Bektas, S., 2014. Shortest distance from a point to triaxial elipsoid. International Journal of Engineering and Applied Sciences, 4(11), 22–26.Search in Google Scholar

[3] Benning, W., 1974. Der kürzeste Abstand eines in rechtwinkligen Koordinaten gegebenen Außenpunktes vom Ellipsoid. Allgemeine Vermessungs-Nachrichten, 81, 429–433.Search in Google Scholar

[4] Bjorck, A. and Dahlquist, G., 1974. Numerical Methods. Prentice-Hall.Search in Google Scholar

[5] Bopp, H. and Krauss, H., 1976. Der Kürzeste Abstand eines Punktes von einem Rotationsellipsoid. Allgemeine Vermessung-Nachrichten, 83, 348–351.Search in Google Scholar

[6] Borkowski, K. M., 1987. Transformation of geocentric to geodetic coordinates without approximations. Astrophys. Space Sci., 139, 1–4.10.1007/BF00643807Search in Google Scholar

[7] Borkowski, K. M., 1989. Accurate algorithms to transform geocentric to geodetic coordinates. Bulletin Géodésique, 63, 50–56.10.1007/BF02520228Search in Google Scholar

[8] Bowring, B. R., 1976. Transformation from spatial to geographical coordinates. Surv. Rev., 23, 323–327.10.1179/sre.1976.23.181.323Search in Google Scholar

[9] Featherstone, W. E. and Claessens, S. J., 2008. Closed-form transformation between geodetic and ellipsoidal coordinates. Stud. Geophys. Geod., 52, 1–18.10.1007/s11200-008-0002-6Search in Google Scholar

[10] Feltens, J., 2009. Vector method to compute the Cartesian (X,Y,Z) to geodetic (φ,λ,h) transformation on a triaxial ellipsoid. J. Geodesy, 83, 129–137. DOI: 10.1007/s00190-008-0246-5.Search in Google Scholar

[11] Fok, H. S. and Iz, B. H., 2003. A comparative analysis of the performance of iterative and non-iterative solutions to the Cartesian to geodetic coordinate transformation. Journal of Geospatial Engineering, 5, 61–74.Search in Google Scholar

[12] Fotiou, A., 1998. A pair of closed expressions to transform geocentric to geodetic coordinates. Zeitschrift für Vermessungswesen, 123, 133–135.Search in Google Scholar

[13] Fukushima, T., 1999. Fast transform from geocentric to geodetic coordinates. J. Geodesy, 73, 603–610.10.1007/s001900050271Search in Google Scholar

[14] Fukushima, T., 2006. Transformation from Cartesian to geodetic coordinates accelerated by Halley’s method. J. Geodesy, 79, 689–693.10.1007/s00190-006-0023-2Search in Google Scholar

[15] Getchell, B. C., 1972. Geodetic latitude and altitude from geocentric coordinates. Celestial Mechanics, 5, 300–302.10.1007/BF01228431Search in Google Scholar

[16] Grafarend, E. W., 2001. Gauss surface normal coordinates in geometry and gravity space. Part 2a: topographic surface normal ellipsoidal coordinates: closed form transformation of “geocentric Cartesian” to “Gauss ellipsoidal coordinates (geodetic)”. Zeitschrift für Vermessungswesen, 126, 373–382.Search in Google Scholar

[17] Guo, G., 2001. A fast procedure for geodetic coordinate transformation. Surveying and Mapping, 11, 12–13.Search in Google Scholar

[18] Heikkinen, M., 1982. Geschlossene Formeln zur Berechnung räumlicher geodätischer Koordinaten aus rechtwinkligen Koordinaten. Zeitschrift für Vermessungswesen, 107, 207–211.Search in Google Scholar

[19] Heiskanen, W. A. and Moritz, H., 1967. Physical Geodesy. Freeman and Co., San Francisco.10.1007/BF02525647Search in Google Scholar

[20] Jones, G. C., 2002. New solutions for the geodetic coordinate transformation. J. Geodesy, 76, 437–446.10.1007/s00190-002-0267-4Search in Google Scholar

[21] Lapaine, M., 1990. A new direct solution of the transformation problem of Cartesian into ellipsoidal coordinates. In: R. H. Rapp and F. Sansò (Eds), Determination of the Geoid: Present and Future, Springer, Berlin, Heidelberg, New York, 395–404.10.1007/978-1-4612-3104-2_46Search in Google Scholar

[22] Lin, K. C. and Wang, J., 1995. Transformation from geocentric to geodetic coordinates using Newton’s iteration. Bulletin Géodésique, 69, 300–303.10.1007/BF00806742Search in Google Scholar

[23] Ligas, M. and Banasik, P., 2011. Conversion between Cartesian and geodetic coordinates on a rotational ellipsoid by solving a system of nonlinear equations. Geodesy and Cartography, 60(2), 145–159. Polish Academy of Sciences.10.2478/v10277-012-0013-xSearch in Google Scholar

[24] Ligas, M., 2013. Various parametrization of “latitude” equation – Cartesian to geodetic coordinates transformation. Journal of Geodetic Science, 3(2), 87–94. DOI: 10.2478/jogs-2013-0012.Search in Google Scholar

[25] Osen, K., 2019. Accurate Conversion of Earth-Fixed Earth-Centered Coordinates to Geodetic Coordinates. HAL Id: hal-01704943, https://hal.archives-ouvertes.fr/hal-01704943v2. Submitted on 31 Oct 2019.Search in Google Scholar

[26] Ozone, M. I., 1985. Non-iterative solution of the ϕ equation. Surveying and Mapping, 45, 169–171.Search in Google Scholar

[27] Panou, G., Korakitis, R. and Delikaraoglou, D., 2018. Cartesian to Geodetic Coordinates Conversion by an Iterative Geometrical Method. In: Quod Erat Demonstrandum – In quest of the ultimate geodetic insight, Special issue for Professor Emeritus Athanasios Dermanis | School of Rural and Surveying Engineering, AUTh, Chapter: 18, Ziti, 274–281.Search in Google Scholar

[28] Paul, M. K., 1973. A note on computation of geodetic coordinates from geocentric (Cartesian) coordinates. Bull. Geod., 108(2), 135–139.10.1007/BF02522075Search in Google Scholar

[29] Pollard, J., 2002. Iterative vector methods for computing geodetic latitude and height from rectangular coordinates. J. Geodesy, 76, 36–40.10.1007/s001900100220Search in Google Scholar

[30] Seemkooei, A. A., 2002. Comparison of different algorithms to transform geocentric to geodetic coordinates. Surv. Rev., 36, 627–633.10.1179/003962602791482966Search in Google Scholar

[31] Sjöberg, L. E., 1999. An efficient iterative solution to transform rectangular geocentric coordinates to geodetic coordinates. Zeitschrift für Vermessungswesen, 124, 295–297.Search in Google Scholar

[32] Sünkel, H., 1976. Ein nicht-iteratives Verfahren zur Transformation geodätischer Koordinaten. Österreichische Zeitschrift für Vermessungswesen, 64, 29–33.Search in Google Scholar

[33] Turner, J. D., 2009. A non-iterative and non-singular perturbation solution for transforming Cartesian to geodetic coordinates. Journal of Geodesy, 83, 139–145. DOI: 10.1007/s00190-008-0247-4.Search in Google Scholar

[34] Vaníček, P. and Krakiwski, E. J., 1982. Geodesy: the concepts. North Holland, Amsterdam, p. 324.Search in Google Scholar

[35] Vermeille, H., 2002. Direct transformation from geocentric to geodetic coordinates. J. Geodesy, 76, 451–454.10.1007/s00190-002-0273-6Search in Google Scholar

[36] Vermeille, H., 2004. Computing geodetic coordinates from geocentric coordinates. J. Geodesy, 78, 94–95.10.1007/s00190-004-0375-4Search in Google Scholar

[37] Wu, Y., Wang, P. and Hu, X., 2003. Algorithm of Earth-centered Earth-fixed coordinates to geodetic coordinates. IEEE Trans. Aerosp. Electron. Syst., 39, 1457–1461.10.1109/TAES.2003.1261144Search in Google Scholar

[38] You, R.-J., 2000. Transformation of Cartesian to geodetic coordinates without iterations. J. Surv. Eng.-ASCE, 126, 1–7.10.1061/(ASCE)0733-9453(2000)126:1(1)Search in Google Scholar

[39] Zeng, H., 2013. Explicitly computing geodetic coordinates from Cartesian coordinates. Earth Planets Space, 65, 291–298.10.5047/eps.2012.09.009Search in Google Scholar

[40] Zhang, C. D., Hsu, H. T., Wu, X. P., Li, S. S., Wang, Q. B., Chai, H. Z. and Du, L., 2005. An alternative algebraic algorithm to transform Cartesian to geodetic coordinates. J. Geodesy, 79, 413–420.10.1007/s00190-005-0487-5Search in Google Scholar

[41] Zhu, J., 1993. Exact conversion of Earth-centered, Earth-fixed coordinates to geodetic coordinates. J. Guid. Control Dyn., 16, 389–391. DOI: 10.2514/3.21016. Published Online: 23 May 2012.Search in Google Scholar

Received: 2020-07-23
Accepted: 2020-10-14
Published Online: 2020-11-04
Published in Print: 2021-01-27

© 2021 Walter de Gruyter GmbH, Berlin/Boston

Articles in the same Issue

  1. Frontmatter
  2. Research Articles
  3. Advantages of tunnel monitoring using distributed fibre optic sensing
  4. Temporal calibration and synchronization of robotic total stations for kinematic multi-sensor-systems
  5. Fusion of ground penetrating radar and laser scanning for infrastructure mapping
  6. Getchell’s method for conversion of Cartesian-geocentric to geodetic coordinates – Its properties and Newtonian alternative
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https://doi.org/10.1515/jag-2020-0034
Keywords for this article
geocentric coordinates; geodetic coordinates; conversion; iterative formula; convergence of algorithms

Articles in the same Issue

  1. Frontmatter
  2. Research Articles
  3. Advantages of tunnel monitoring using distributed fibre optic sensing
  4. Temporal calibration and synchronization of robotic total stations for kinematic multi-sensor-systems
  5. Fusion of ground penetrating radar and laser scanning for infrastructure mapping
  6. Getchell’s method for conversion of Cartesian-geocentric to geodetic coordinates – Its properties and Newtonian alternative
  7. Characteristics of seasonal variations and noises of the daily double-difference and PPP solutions
  8. Evaluation of multipath mitigation performance using signal-to-noise ratio (SNR) based signal selection methods
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