Skip to main content
SpringerLink
Log in

Analytical Approximation of Functional Dependences of the Geodesic Line Parameters

  • Published:
Mechanics of Solids Aims and scope Submit manuscript

Abstract

The synthetization of high-precision analytical approximations of the geodesic longitude dependence on the reduced latitude on the spheroid geodesic line is considered. The obtained approximations are relevant for both geodetic and navigation problems providing the capability of significant reduction in computational costs in high-precision geodetic and navigation calculations.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1.
Fig. 2.

Similar content being viewed by others

REFERENCES

  1. B. B. Serapinas, Geodesic Basics of Maps (MSU, Moscow, 2001) [in Russian].

    Google Scholar 

  2. Wei-Kuo Tseng, M. A. Earle, and Jiunn-Liang Guo, “Direct and inverse solutions with geodetic latitude in terms of longitude for rhumb line sailing,” J. Navig. 65 (3), 549–559 (2012).

    Article  Google Scholar 

  3. N. E. Daidzic, “Long and short-range air navigation on spherical Earth,” Int. J. Aviat., Aeronaut. Aerospace 4 (1), 1–54 (2017).

    Google Scholar 

  4. S. V. Sokolov, “Analytical models of spatial trajectories for solving navigation problems,” J. Appl. Math. Mech. 79 (1), 17–22 (2015).

    Article  MathSciNet  Google Scholar 

  5. I. N. Rozenberg, S. V. Sokolov, V. I. Umanskii, and V. A. Pogorelov, Theoretical Foundations of Close Integration of Inertial-Satellite Navigation Systems (Fizmatlit, Moscow, 2018) [in Russian].

    Google Scholar 

  6. P. A. Kucherenko and S. V. Sokolov, “Analytical Solution for a Problem on Approximation of Functional Dependences for Parameters of a Geodesic Line,” Mech. Solids 54 (7), 1076–1083 (2019).

    Article  ADS  Google Scholar 

  7. V. P. Morozov, Course of Spheroidal Geodesy (Nedra, Moscow, 1979) [in Russian].

    Google Scholar 

  8. I. S. Gradshtein and I. M. Ryzhik, Tables of Integrals (Fizmatlit, Moscow, 1964) [in Russian].

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Rostov State Transport University, 344038, Rostov-on-Don, Russia

    P. A. Kucherenko

  2. Moscow Technical University of Communications and Informatics, 123423, Moscow, Russia

    S. V. Sokolov

Authors
  1. P. A. Kucherenko
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. S. V. Sokolov
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding authors

Correspondence to P. A. Kucherenko or S. V. Sokolov.

Additional information

Translated by E. Oborin

Appendices

APPENDIX A

The coefficients of series expansion of the function \(\sqrt {1 - {{e}^{2}}{{{\cos }}^{2}}u} \) are as follows

$${{\alpha }_{0}} = \sqrt {1 - {{e}^{2}}{{{\cos }}^{2}}{{u}_{l}}} + \frac{{{{e}^{2}}{{{\cos }}^{2}}{{u}_{l}}}}{{2\sqrt {1 - {{e}^{2}}{{{\cos }}^{2}}{{u}_{l}}} }} - \frac{{{{e}^{4}}{{{\cos }}^{4}}{{u}_{l}}}}{{8\sqrt {{{{(1 - {{e}^{2}}{{{\cos }}^{2}}{{u}_{l}})}}^{3}}} }}$$
$$ + \;\frac{{{{e}^{6}}{{{\cos }}^{6}}{{u}_{l}}}}{{16\sqrt {{{{(1 - {{e}^{2}}{{{\cos }}^{2}}{{u}_{l}})}}^{5}}} }} - \frac{{5{{e}^{8}}{{{\cos }}^{8}}{{u}_{l}}}}{{128\sqrt {{{{(1 - {{e}^{2}}{{{\cos }}^{2}}{{u}_{l}})}}^{7}}} }}$$
$${{\alpha }_{1}} = - \frac{{{{e}^{2}}}}{{2\sqrt {1 - {{e}^{2}}{{{\cos }}^{2}}{{u}_{l}}} }} + \frac{{{{e}^{4}}{{{\cos }}^{2}}{{u}_{l}}}}{{4\sqrt {{{{(1 - {{e}^{2}}{{{\cos }}^{2}}{{u}_{l}})}}^{3}}} }}$$
$$ - \;\frac{{3{{e}^{6}}{{{\cos }}^{4}}{{u}_{l}}}}{{16\sqrt {{{{(1 - {{e}^{2}}{{{\cos }}^{2}}{{u}_{l}})}}^{5}}} }} + \frac{{5{{e}^{8}}{{{\cos }}^{6}}{{u}_{l}}}}{{32\sqrt {{{{(1 - {{e}^{2}}{{{\cos }}^{2}}{{u}_{l}})}}^{7}}} }}$$
$${{\alpha }_{2}} = - \frac{{{{e}^{4}}}}{{8\sqrt {{{{(1 - {{e}^{2}}{{{\cos }}^{2}}{{u}_{l}})}}^{3}}} }} + \frac{{3{{e}^{6}}{{{\cos }}^{2}}{{u}_{l}}}}{{16\sqrt {{{{(1 - {{e}^{2}}{{{\cos }}^{2}}{{u}_{l}})}}^{5}}} }} - \frac{{15{{e}^{8}}{{{\cos }}^{4}}{{u}_{l}}}}{{64\sqrt {{{{(1 - {{e}^{2}}{{{\cos }}^{2}}{{u}_{l}})}}^{7}}} }}$$
$${{\alpha }_{3}} = - \frac{{{{e}^{6}}}}{{16\sqrt {{{{(1 - {{e}^{2}}{{{\cos }}^{2}}{{u}_{l}})}}^{5}}} }} + \frac{{5{{e}^{8}}{{{\cos }}^{2}}{{u}_{l}}}}{{32\sqrt {{{{(1 - {{e}^{2}}{{{\cos }}^{2}}{{u}_{l}})}}^{7}}} }}$$
$${{\alpha }_{4}} = - \frac{{5{{e}^{8}}}}{{128\sqrt {{{{(1 - {{e}^{2}}{{{\cos }}^{2}}{{u}_{l}})}}^{7}}} }}.$$

APPENDIX B

The expressions for the expansion coefficients \({{\beta }_{n}}\) are as follows

$${{\beta }_{0}} = \phi \frac{{24{{\alpha }_{2}}{{k}^{4}} + 6{{\alpha }_{3}}(8{{k}^{2}} - 3){{k}^{2}} + {{\alpha }_{4}}(72{{k}^{4}} - 54{{k}^{2}} + 15)}}{{48{{k}^{5}}}}$$
$${{\beta }_{1}} = - \phi \frac{{{{\alpha }_{4}}(18{{k}^{2}} - 5) + 6{{k}^{2}}{{\alpha }_{3}}}}{{24{{k}^{3}}}}$$
$${{\beta }_{2}} = \phi \frac{{{{\alpha }_{4}}}}{{6k}}$$
$${{\beta }_{3}} = \phi \frac{{{{\alpha }_{2}}(2{{k}^{2}} - 1)8{{k}^{4}} + {{\alpha }_{3}}(8{{k}^{4}} - 8{{k}^{2}} + 3)2{{k}^{2}} + {{\alpha }_{4}}(16{{k}^{6}} - 24{{k}^{4}} + 18{{k}^{2}} - 5)}}{{16{{k}^{6}}}}$$
$${{\beta }_{4}} = - ({{\beta }_{0}} + {{\beta }_{1}}{{\sin }^{2}}{{u}_{0}} + {{\beta }_{2}}{{\sin }^{4}}{{u}_{0}})\sin {{u}_{0}}\sqrt {1 - {{k}^{2}}{{{\sin }}^{2}}{{u}_{0}}} - {{\beta }_{3}}\arcsin (k\sin {{u}_{0}})$$

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Kucherenko, P.A., Sokolov, S.V. Analytical Approximation of Functional Dependences of the Geodesic Line Parameters. Mech. Solids 55, 1210–1215 (2020). https://doi.org/10.3103/S0025654420080130

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.3103/S0025654420080130

Keywords:

Search

Navigation