The dual of sweep

Abstract

As an infinite 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 article formally defines 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 difference of E and the trajectory generated by the inverted motion M̂. In this sense, the operation of unsweep is a generalization of Minkowski difference that corresponds to a ‘material removal’ operation, it can be also defined as an infinite 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. Using duality, the established properties of unsweep is extended to the general sweep operation, and can be used to improve the computational support for general sweeps.