Given a collection of regions on a map, we seek a method of continuously altering the regions as the scale is varied. This is formalized and brought to rigor as well-defined problems in homotopic deformation. We ask the regions to preserve topology, area-ratios, and relative position as they change over time. A solution is presented using differential methods and computational geometric techniques. Most notably, an application of this method is used to provide an algorithm to obtain cartograms.