Transitions between 4-intersection values of planar regions

Abstract

If A(t) and B(t) are subsets of the Euclidean plane which are continuously morphing, we investigate the question of whether they may morph directly from being disjoint to overlapping so that the boundary and interior of A(t) both intersect the boundary and interior of B(t) without first passing through a state in which only their boundaries intersect.  More generally, we consider which 4-intersection values---binary 4-tuples specifying whether the boundary and interior of A(t) intersect the boundary and interior of B(t)---are adjacent to which in the sense that one may morph into the other without passing through a third value.  The answers depend on what forms the regions A(t) and B(t) are allowed to assume and on the definition of continuous morphing of the sets.