Reconfiguring closed polygonal chains in Euclidean d-space

Abstract

Consider the problem of moving a closed chain of n links in two or more dimensions from one given configuration to another. The links have fixed lengths and may rotate about their endpoints, possibly passing through one another. The notion of a "line-tracking motion" is defined, and it is shown that when reconfiguration is possible by any means, it can be achieved by O(n) line-tracking motions. These motions can be computed in O(n) time on real RAM. It is shown that in three or more dimensions, reconfiguration is always possible, but that in dimension two this is not the case. Reconfiguration is shown to be always possible in two dimensions if and only if the sum of the lengths of the second and third longest links add to at most the sum of the lengths of the remaining links. An O(n) algorithm is given for determining whether it is possible to move between two given configurations of a closed chain in the plane and, if it is possible, for computing a sequence of line-tracking motions to carry out the reconfiguration. © 1995 Springer-Verlag New York Inc.