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The class of problems that involve finding where to place or how to move a solid object in the presence of obstacles is discussed. The solution to this class of problems is essential to the automatic planning of manipulator transfer movements, i.e., the motions to grasp a part and place it at some destination. For example, planning transfer movements requires the ability to plan paths for the manipulator that avoid collisions with objects in the workspace and the ability to choose safe grasp points on objects. The approach to these problems described here is based on a method of computing an explicit representation of the manipulator configurations that would bring about a collision.