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The development of technologies in the field of geodesy and map projections is important for the coordinate system used in surveying works and geographic information system \"GIS\". This paper presents a system of coordinates by harmonic equations projection \"the united projections\" that has five projections (Mercator, Lambert, Russell, Lagrange, and the compound projections) in one zone coordinate system. The theory of the projections by a harmonic equation as well as Lagrange projection has eight direct algorithms defined by Professor Vladimir podshivolev 1998. These algorithms have some difficulties and very complicated method. A new direct algorithms for all five projections have been presented, as well as a new coordinate system by compound projection for Scandinavian countries. Also the distortion scale factor for measuring distances in Sweden by the smart main system for some cities have been discussed.