Direct Expansions of Transformations Between Three Kinds of Latitudes Used in Map Projection

Considering the traditional transformations between rectifying,conformal and authalic latitudes are comparatively fuzzy,the direct expansions of transformations between three kinds of latitudes are comprehensively derived with the help of computer algebra system Mathematica using the forward and inverse expansions of transformations between the three kinds of latitudes and geodetic latitude.Their coefficients are expressed in a power series of the ellipsoid's first eccentricity,e,and extended up to tenth-order terms of e,which could solve the transformation problem when different reference ellipsoids are used.Numerical examples show that the accuracies of these direct expansions are higher than 10-8″,and could completely meet the needs of precise calculation in map projection.