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

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 mp (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.

Comparison of main geometric characteristics of deformed sphere and standard spheroid

“…It turns out that the geodesics on λ -spheres can be expressed through the well-known analytical functions (inverse tangent), whereas the geodesics on the spheroids are expressed through the incomplete elliptic integrals of the first and third kind (see Section 3). The above observation justifies our idea to propose a new reference model for the geoid (i.e., the surface that approximates the actual shape of the Earth [6]) that is based on the λ -spheres (see [4,7,8]) alternatively to the standard reference models that are based on the rotational ellipsoids.…”

“…where φ ∈ [0, π] and v ∈ [−π, π] are the latitude and longitude variables on the spheroid respectively. We can also define the first and second eccentricities and the flattening factor describing the spheroid (7) as…”

“…Let us identify the semi-major and semi-minor axes a and b of the spheroid (7) with the equatorial radius R and the semiheight H λ of the λ -sphere at the level of the North Pole (or, in other words, the polar radius of the λ -sphere), i.e., R ≡ a and…”

λ‐spheres as a new reference model for the geoid

“…As for the future research directions, it is planned to investigate also the physical aspects of the newly proposed $$$ \lambda $$$‐sphere's reference model for the geoid, for example, the determination of the theoretical gravity reference field at any position on, above, or below the equipotential surface of the $$$ \lambda $$$‐sphere. Additionally, we are going to analyze in detail in our following papers the direct (when the starting point on the $$$ \lambda $$$‐sphere and the geodesic are known and we need to find the ending point) and inverse (when the starting and ending points are known and we need to find the geodesic connecting those two points) problems for geodesics and loxodromes (rhumb lines) 15 for some practical calculations in geodesy and navigation on the Earth's surface. For the inverse geodesic problems, it is also important to study the existence of the conjugate (or focal) points on the $$$ \lambda $$$‐sphere's surface.…”

“…•) based on the metric tensor g defined in the manifold M. Therefore, all geodesics, that is, the shortest paths that locally minimize the distance between any two given points 𝛼(a) = p and 𝛼(b) = q are defined as the minima of the above variational problem (15).…”