Shape deformation in continuous map generalization

Given a collection of regions on a map, we seek a method of altering the regions over time so they are still visible when the map is scaled. 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 conjectured, using differential methods, leading to possible applications to cartograms. The problem is also extended to include the notion of epsilon-separation, which addresses issues related to coalescence, self-coalescence, and imperceptibility.