The accurate calculation of the geodetic heights of the celestial body’s surface points relative to the triaxial ellipsoid

A sphere or ellipsoid of revolution are usually used for approximation of the physical surface of the Earth. In some cases, a triaxial ellipsoid is used. The calculation of the geodetic height of points on the Earth’s surface is carried out mainly by approximate methods using the formulas for the dependence of spatial rectangular coordinates x, y, z on geodesics B, L, H. However, there are small bodies of the Solar system, for example, Eros 433 asteroid, for which such variants of the first approximation are incorrect, since in this case both first approximations are not small quantities. This article proposes a fundamentally new approach to calculating the geodesic height relative to the triaxial ellipsoid, based on the joint use of the normal equation for a surface passing through a given point and the equations of the surface itself. The method is reduced to solving the sixth-degree equation by the Sturm method and the fourth-degree equation by the Ferrari method.