Comparison of main geometric characteristics of deformed sphere and standard spheroid

Abstract

In the paper we compare the geometric descriptions of the deformed sphere (i.e., the so-called λ -sphere) and the standard spheroid (namely, World Geodetic System 1984's reference ellipsoid of revolution). Among the main geometric characteristics of those two surfaces of revolution embedded into the three-dimensional Euclidean space we consider the semi-major (equatorial) and semi-minor (polar) axes, quartermeridian length, surface area, volume, sphericity index, and tipping (bifurcation) point for geodesics. Next, the RMS (Root Mean Square) error is defined as the square-rooted arithmetic mean of the squared relative errors for the individual pairs of the discussed six main geometric characteristics. As a result of the process of minimization of the RMS error, we have obtained the proposition of the optimized value of the deformation parameter of the λ -sphere, for which we have calculated the absolute and relative errors for the individual pairs of the discussed main geometric characteristics of λ -sphere and standard spheroid (the relative errors are of the order of 10 −6 -10 −9 ). Among others, it turns out that the value of the flattening factor of the spheroid is quite a good approximation for the corresponding value of the deformation parameter of the λ -sphere (the relative error is of the order of 10 −4 ).