An iterative approach for inverse transformation of map projections

ABSTRACT

Map projections are given by forward transformation equations. Inverse transformation is derived from forward transformation analytically or numerically. In this paper, a numerical approach for inverse transformation of map projections is proposed, which is based on numerical differentiation and Newton–Raphson root finding method. This approach can facilitate the program developments for map projections when inverse transformation is needed. Numerical differentiation is tested with three map projections. It is seen that seven-digit precision or more can be reached. Boundary conditions and initial guess problem in inverse transformation are discussed. In terms of initial guess, map projections are divided into three categories, and appropriate initial guess values for cylindrical, pseudocylindrical, azimuthal, and conical projections in normal aspect are suggested. Newton–Raphson method with numerical differentiation is tested with 20 different map projections by using test data sets. The results show that the proposed approach is applicable if appropriate initial guess is available.