In this paper we present tight bounds on the maximum size of aspect graphs and give worstcase optimal algorithms for their construction, first in the convex case and then in the general case. In particular, we give upper and lower bounds on the maximum size (including vertex labels) of Θ(n3) and Θ(n5) and algorithms for constructing the aspect graph which run in time O(n3) and O(n5) for the convex and general cases respectively. The algorithm for the general case makes use of a new 3D object representation called the aspect representation or asp. We also show a different way to label the aspect graph in order to save a factor of n in the asymptotic size (at the expense of label retrieval time) in both the convex and general cases, and we suggest alternatives to the aspect graph which require less space and store more information.

In this paper we consider from a theoretical viewpoint the complexity of some reachability and motion planning questions. Specifically, we are interested in determining which generalizations of the basic mover's problem result in computationally intractable problems. It has been shown that for any set of motion-planning problems with bounded degree of freedom, there is a polynomial-time algorithm to solve the motion-planning problem (although the degree of the polynomial may be large), but the two most basic generalizations to the problem, multiple movable obstacles and conformable objects, result in much harder problems. It has been shown that the warehouseman's problem is P-space hard: in this paper we show that the reachability problem for one of the simplest types of conformable objects, a two-dimensional linear (“robot arm”) linkage, is P-space complete. In addition, we demonstrate some motion-planning problems that take exponential time.