The dual of sweep

As an in nite union operation sweep of a moving object through space is a powerful and natural addition to the Boolean set operations that incorporates motion related information for the purposes of shaping collision detection and simulation of moving objects Use of sweep has been hindered by limited computational support and by the fact that it is a material growing operation whereas many applications such as interference elimination and mechanism design appear to be better modeled by a material removal operation This paper formally de nes a new geometric modeling operation of unsweep Given an arbitrary subset E of Euclidean space and a general motion M unsweep E M returns the largest subset of E that remains inside the original E under M When M is a translation unsweep E M naturally reduces to the standard Minkowski di erence of E and the trajectory generated by the inverted motion M In this sense the operation of unsweep is a generalization of Minkowski di erence that corresponds to a material removal operation it can be also de ned as an in nite intersection operation and is the dual of sweep in a precise set theoretic sense We show that unsweep has attractive theoretical and computational properties including a practical point membership test for arbitrary general motions By duality the established properties of unsweep extend to the general sweep operation and can be used to improve the computational support for general sweeps