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. 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 the boundary function; the transition graph between RCC
relations (Randell, Cui, and Cohn, 1992). We discuss the physical
significance of these different criteria.
We also show that the history-based definition of continuity proposed
by Muller (1998) is equivalent to continuity with respect to the
Hausdorff distance. An examination of the difference between the
transition rules that we have found for the Hausdorff distance and the
transition theorems that Muller derives leads to the conclusion that
Muller's analysis of state transitions is not adequate. We propose an
alternative characterization of transitions in Muller's first-order
language over histories.