TABLE OF CONTENTS In the previous section we introduced vector fields for high-level control of micro actuator arrays. In particular, it was stated that in a squeeze field, polygonal parts have a (usually very small) finite number of orientation equilibria (see Claim 4). Because of this result, squeeze fields play a key role in manipulation strategies. In this section, we survey recent research on the combinatorial, geometric, and algorithmic properties of squeeze fields, by analyzing the area bisectors of a part. The results in Sec. 6 were obtained in collaboration with Danny Halperin [14, 15, 16].
There is a direct relationship between equilibria in squeeze fields and area bisectors. Recall Figure 15: a part is in force equilibrium if and only if the squeeze line bisects the part into two sections of equal area.
[14] Let P be a polygon in the plane, possibly with holes, and having n vertices in total. We denote by V the set of vertices of P. For a directed line ? in the plane, we denote by h l( ?) (resp. h r( ?)) the open half-plane bounded by ? on the left- (resp. right-) hand-side of ?. The line ? is an area bisector of P if the area of P ? h l( ?) is equal to the area of P ? h r( ?).
A line ?
