Inverse task of recalculating coordinates from the projection plane onto a triaxial ellipsoid

Using projections of the triaxial ellipsoid in GIS technologies it is necessary to solve problems of recalculating coordinates from a geographic into a rectangular system (direct task) and from planar rectangular coordinates into those geographic (inverse one) on the approximation surface. The method of recalculating them is developed on the example of a cylindrical projection of a meridian section, that azimuthal, preserving length along the meridians, and the Jacobi one. The list of projections is determined by the previously obtained equations and the necessity to cover their different classes. To demonstrate and confirm the correctness of the calculations, cartographic grids and photo maps are made for a reference surface approximating the shape of Phobos, a satellite of Mars. In general, it is impossible to determine the values of arguments from a given value of two variables’ function. For the listed projections, the solution can be reduced to a sequential determination of the argument’s value from that of one variable’s function using numerical techniques, in particular, the method of dividing a segment in half