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.

@article{danciger09shape,
  Author = {Jeff Danciger and Satyan Devadoss and John Mugno and Donald R. Sheehy and Rachel Ward},
  Journal = {GeoInformatica},
  Number = {2},
  Pages = {203--221},
  Title = {Shape Deformation in Continuous Map Generalization},
  Volume = {13},
  Year = {2009}}