Symbolic Expressions for Direct and Inverse Solutions to Common Latitudes with the Variable of Geocentric Latitude

  Objectives  For traditional ellipsoid geometry, the transformations of direct and inverse solutions to common latitudes are carried out with the geodetic latitude as the variable, which is complex for understanding. However, the geocentric latitude is more intuitive than the geodetic latitude. Therefore, the direct and inverse solutions to common latitudes with the geocentric latitude as the variable are derived, which is not only the supplementary study of the latitude transformation theory but also the optimization of the traditional latitude transformation method.

  Methods  Firstly, the rectifying latitude, conformal latitude, and authalic latitude are expressed as the functions of geocentric latitude. Secondly, with the help of the powerful computer algebra system Mathematica, the coefficients are expressed in the power series of the elliptical eccentricity e and the third flattening rate n of the ellipsoid in the formulas respectively, and the direct solution expressions are obtained. On this basis, the inverse solution expressions are derived by the symbolic iteration method. Finally, through the example of CGCS2000 ellipsoid parameters, the accuracy of the derived formulas is verified with the maximum difference as the standard.

  Results  Numerical examples results show that the calculation error of the expressions based on e is less than $\left(\mathrm{1}\times {\mathrm{10}}^{-\mathrm{8}}\right)″$, and that based on n is less than $\left(\mathrm{1}\times {\mathrm{10}}^{-\mathrm{10}}\right)″$, which can completely meet the accuracy requirement of geodesy and map projection.

  Conclusions  Compared with the expansions of the auxiliary latitude with the geodetic latitude as the variable, the expansions of common latitudes with the geocentric latitude as the variable are consistent with them in structures and forms. Theoretically, the geocentric latitude can be used as a supplementary means of auxiliary latitude transformations, and the power series expressions based on n have more compact forms, better convergence, and higher calculation accuracy.