Expressions of Analytical Transformations Between Lagrange and the Commonly Used Conformal Projections by Complex Numbers

The problem of the analytical transformations between Lagrange and the commonly used conformal projections is discussed with the help of complex numbers theory.The forward and inverse expressions of Lagrange projection by complex numbers are derived,and the mathematical modes of the analytical transformations between Lagrange and Gauss,Mercator,Lambert conformal conic projections by complex numbers are systematically established.The formulae derived by the paper are in the symbolic form including the first eccentricity of the reference ellipsoid,so they could solve the transformation problems when different reference ellipsoids are used.Compared to traditional transformation formulae in the real number domain,they have more concise structure and stricter theory basis,and are more convenient for practical use.