λ -Spheres as a New Reference Model for Geoid: Explicit Solutions of the Direct and Inverse Problems for Loxodromes (Rhumb Lines)

Vasyl Kovalchuk () and Ivaïlo M. Mladenov
Additional contact information
Vasyl Kovalchuk: Institute of Fundamental Technological Research, Polish Academy of Sciences, 5B, Pawińskiego Str., 02-106 Warsaw, Poland
Ivaïlo M. Mladenov: Institute for Nuclear Research and Nuclear Energy, Bulgarian Academy of Sciences, Tsarigradsko Chaussee 72, 1784 Sofia, Bulgaria

Mathematics, 2022, vol. 10, issue 18, 1-10

Abstract: In this paper, we present a new reference model that approximates the actual shape of the Earth, based on the concept of the deformed spheres with the deformation parameter λ . These surfaces, which are called λ -spheres, were introduced in another setting by Faridi and Schucking as an alternative to the spheroids (i.e., ellipsoids of revolution). Using their explicit parametrizations that we have derived in our previous papers, here we have defined the corresponding isothermal (conformal) coordinates as well as obtained and solved the differential equation describing the loxodromes (or rhumb lines) on such surfaces. Next, the direct and inverse problems for loxodromes have been formulated and the explicit solutions for azimuths and arc lengths have been presented. Using these explicit solutions, we have assessed the value of the deformation parameter λ for our reference model on the basis of the values for the semi-major axis of the Earth a and the quarter-meridian m p (i.e., the distance between the Equator and the North or South Pole) for the current best ellipsoidal reference model for the geoid, i.e., WGS 84 (World Geodetic System 1984). The latter is designed for use as the reference system for the GPS (Global Positioning System). Finally, we have compared the results obtained with the use of the newly proposed reference model for the geoid with the corresponding results for the ellipsoidal (WGS 84) and spherical reference models used in the literature.

Keywords: deformed spheres; incomplete elliptic integrals; geoid’s reference models; loxodromes or rhumb lines; azimuths and arc lengths; geodesy and navigation problems (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2022
References: View complete reference list from CitEc
Citations:

Downloads: (external link)
https://www.mdpi.com/2227-7390/10/18/3356/pdf (application/pdf)
https://www.mdpi.com/2227-7390/10/18/3356/ (text/html)

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:gam:jmathe:v:10:y:2022:i:18:p:3356-:d:916038

Access Statistics for this article

Mathematics is currently edited by Ms. Emma He

More articles in Mathematics from MDPI
Bibliographic data for series maintained by MDPI Indexing Manager ().