Continuous shape transformation and metrics on regions

A natural approach to defining continuous change of shape is in terms of a metric that measures the difference between two regions. We consider four such metrics over regions: the Hausdorff distance, the dual-Hausdorff distance, the area of the symmetric difference, and the optimal-homeomorphism metric (a generalization of the Fréchet distance). Each of these gives a different criterion for continuous change. We establish qualitative properties of all of these; in particular, the continuity of basic functions such as union, intersection, set difference, area, distance, and smoothed circumference and the transition graph between RCC-8 relations. We also show that the history-based definition of continuity proposed by Muller is equivalent to continuity with respect to the Hausdorff distance.