On determination the relationship of geodesic parameters using the theory of implicit functions

An implicit function can be represented only in numerical format. For example, the figure of the geoid of the Earth as a surface of equal potentials in the final version is provided by a set of numerical data, based on which its graphical representation is built. Such a presentation does not allow analytical studies of surface features. This situation also occurs in other cases of presenting the function in the implicit form. Determining the analytical form of the geoid shape is the main task of the theory of the shape of the Earth. At the same time, the theory of implicit functions makes it possible to propose another method of establishing an analytical representation of a function given in the implicit form. The use of Taylor’s formula for functions of multiple variables with a preliminary definition of the derivatives for the function being studied gives the necessary representation. At the same time, the accuracy of the approximation is controlled by the number of members of the Taylor series, and the derivatives themselves are calculated analytically using mathematical software packages. The proposed representation algorithm is valid both for one and for a set of functions determined from the system of equalities.