We discuss the nature of geostrophic momentum approximation as the basis of a semi-geostrophic (SG) model capable of simulating fronts. We formulate the governing equations of a 2D SG model, particularly the SG tendency equation and Sawyer-Eliassen (SE) equation. We examine a canonical solution of the SE equation that reveals the dependence of a frontal transverse circulation upon the three key parameters. We go over the algorithm of performing a complete analysis of a SG model in physical coordinates. We derive two coupled diagnostic equations for the transverse circulation in a 3D SG model. They are a generalization of the SE equation.