Fitting a triaxial ellipsoid to a geoid model

Abstract

The aim of this work is the determination of the parameters of Earth’s triaxiality through a geometric fitting of a triaxial ellipsoid to a set of given points in space, as they are derived from a geoid model. Starting from a Cartesian equation of an ellipsoid in an arbitrary reference system, we develop a transformation of its coefficients into the coordinates of the ellipsoid center, of the three rotation angles and the three ellipsoid semi-axes. Furthermore, we present different mathematical models for some special and degenerate cases of the triaxial ellipsoid. We also present the required mathematical background of the theory of least-squares, under the condition of minimization of the sum of squares of geoid heights. Also, we describe a method for the determination of the foot points of the set of given space points. Then, we prepare suitable data sets and we derive results for various geoid models, which were proposed in the last fifty years. Among the results, we report the semi-axes of the triaxial ellipsoid of geometric fitting with four unknowns to be 6378171.92 m, 6378102.06 m and 6356752.17 m and the equatorial longitude of the major semi-axis –14.9367 degrees. Also, the parameters of Earth’s triaxiality are directly estimated from the spherical harmonic coefficients of degree and order two. Finally, the results indicate that the geoid heights with reference to the triaxial ellipsoid are smaller than those with reference to the oblate spheroid and the improvement in the corresponding rms value is about 20 per cent.

A triaxial reference ellipsoid for the Earth

“…The geoid undulations with respect to the triaxial reference ellipsoid are presented. We also discuss our results in comparison with a least-squares solution by Panou et al (2020).…”

“…Earth's triaxiality has been investigated historically via astro-geodetic or gravimetric measurements (Clarke 1861; Heiskanen 1928), and since the advent of the space era, (together with) observed satellite motions (Kaula 1959;Izsak 1961;Kozai 1961). As an approximate equipotential surface, this reference figure, whether a biaxial or triaxial ellipsoid, can be determined as best fits to the geoid derived from the gravitational field model (Burša 1970;Burša and Sima 1980;Tserklevych et al 2016;Panou et al 2020;Soler and Han 2020). This is the same principle as determining the triaxial dimensions of other planetary bodies (Smith et al 1999;Iz et al 2011).…”

“…Granted, a biaxial ellipsoid remains an intuitive and apposite reference for the Earth in near hydrostatic equilibrium. With the measurement precision nowadays far exceeding the (in)distinctness of the equatorial flattening, the triaxial ellipsoid has received renewed attention for being a more accurate and natural reference figure (Panou et al 2020;Soler and Han 2020). This work exploits the innate advantages of the EHs and is among the first to offer a practical solution on the topic.…”

“…It is informative to compare the result with another solution by Panou et al (2020). They derived the reference ellipsoids for various gravitational field models via a geometric method, i.e., as the best-fitting shapes to the corresponding geoid models.…”

The shape of aroma: measuring and modeling citrus oil gland distribution

“…Each fruit is defined by a point cloud made by the centers of all its individual oil glands. The parameters of the best-fit ellipsoid for this point cloud are computed following the algorithm by Li and Griffiths (2004), from which the semi-axes lengths, rotations, and center are determined (Panou et al, 2020). The fruit point cloud is then rotated and translated such that the best-fit ellipsoid is centered at the origin and its semi-major axes coincide with the proximal-distal axis of the fruit.…”