Coordinate transformations refer to mathematical processing that enables overlay of digital maps that use different coordinate reference systems, that is, map projections. The transformation from geographical to map (plane) coordinates is the conventional practice in cartography, which is called forward transformation. The inverse transformation. which yields geographical coordinates from map coordinates, is a more recent development. due to the need for transformation between different map projections, e specially in geographic information systems (GIS). The combination of the inverse and forward transformation from one projection to another. which may be called grid-on-grid or map-to-map transformation. can be necessary for some custom applications in GIS and in automated cartography Many different approximation algorithms can be used for this problem on desktop computers. In this paper a new local transformation method called a non-Sibsonian transformation. which uses non-Sibson local coordinates. is suggested for map-m-map transformation or for improving geometrical accuracy of scanned maps. A case Study is performed using the Lambert conformal conic projection and is presented with results.