The Direct Calculating Formulae for Transformations Between Authalic Latitude Function and Isometric Latitude

In order to realize the direct transformations between authalic and conformal projections,the direct calculating formulae for transformations between authalic latitude function and isometric latitude are derived with the help of computer algebra system Mathematica.Their coefficients are expressed uniformly in a power series of the first eccentricity of the ellipsoid,and the transformation problem when different reference ellipsoids are used could be solved.Numerical examples show that errors of the two formulae is less than 10 m2 and 10-4″respectively,which could satisfy practical application.